数学中国

标题: 如何把 1/7 拆分成四个不同单位分数的和，并且四个分母之和为最小？ [打印本页]

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作者: 天山草    时间: 2026-1-9 19:15
标题: 如何把 1/7 拆分成四个不同单位分数的和，并且四个分母之和为最小？
如何把 1/7 拆分成四个不同单位分数的和，并且四个分母之和为最小？

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作者: 天山草    时间: 2026-1-10 09:30
1/7 = 1/8 + 1/120 + 1/154 + 1/330;
上式正确，但是右边分母之和是 612，并不是最小的。

1/7 == 1/14 + 1/28 + 1/42 + 1/84;
上式正确，右边分母之和是 168，猜想这才是最小的。

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作者: 天山草    时间: 2026-1-10 09:34
上楼中分母之和最小的等式是如何得到的？见下面：
1/7 = 1/7 (1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4)
也就是：
1/7 = 1/7 (1/2 + 1/6 + 1/12 + 1/4)
即：
1/7 = 1/14 + 1/42 + 1/84 + 1/28

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作者: tmduser    时间: 2026-1-10 12:25


猜想错误
1/20 + 1/28 +1/30 + 1/42 = 1/7
20 + 28 + 30 + 42 = 120

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作者: 天山草    时间: 2026-1-10 18:25

tmduser 发表于 2026-1-10 12:25
猜想错误
1/20 + 1/28 +1/30 + 1/42 = 1/7
20 + 28 + 30 + 42 = 120

看来这个问题并不简单。不知道还有没有分母之和比 120 更小的？

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作者: 王守恩    时间: 2026-1-14 11:50
{{6,  99, {20, 21, 28, 30}},  {7, 120, {21, 24, 35, 40}},     {8, 135, {24, 30, 36, 45}},      {9, 153, {26, 36, 39, 52}},     {10, 168, {30, 40, 42, 56}},
{11, 180, {36, 44, 45, 55}}, {12, 198, {40, 42, 56, 60}},    {13, 216, {39, 52, 60, 65}},     {14, 229, {45, 55, 63, 66}},    {15, 246, {50, 55, 66, 75}},
{16, 261, {55, 60, 66, 80}},  {17, 308, {44, 68, 77, 119}},    {18, 297, {60, 63, 84, 90}},     {19, 330, {55, 66, 95, 114}},    {20, 323, {70, 78, 84, 91}},
{21, 344, {70, 78, 91, 105}},   {22, 360, {72, 88, 90, 110}},    {23, 405, {60, 92, 115, 138}},  {24, 387, {88, 90, 99, 110}},    {25, 417, {75, 100, 110, 132}},
{26, 420, {91, 104, 105, 120}}, {27, 440, {90, 108, 110, 132}}, {28, 458, {90, 110, 126, 132}}, {29, 546, {78, 91, 174, 203}},   {30, 485, {108, 110, 132, 135}},
{31, 626, {72, 120, 155, 279}}, {32, 522, {110, 120, 132, 160}},{33, 536, {112, 126, 144, 154}}, {34, 583, {117, 119, 126, 221}},{35, 576, {112, 140, 144, 180}},
{36, 583, {126, 136, 153, 168}}, {37, 713, {84, 148, 222, 259}}, {38, 612, {136, 152, 153, 171}}, {39, 641, {136, 144, 153, 208}}, {40, 646, {140, 156, 168, 182}},
{41, 776, {120, 123, 205, 328}}, {42, 681, {147, 156, 182, 196}}, {43, 906, {129, 132, 172, 473}}, {44, 706, {165, 171, 180, 190}}, {45, 723, {168, 175, 180, 200}},
{46, 775, {156, 161, 182, 276}}, {47, 1008, {112, 144, 329, 423}}, {48, 774, {176, 180, 198, 220}}, {49, 824, {147, 180, 245, 252}}, {50, 817, {175, 180, 210, 252}},
{51, 839, {176, 187, 204, 272}},  {52, 840, {182, 208, 210, 240}},  {53, 1189, {120, 168, 371, 530}}, {54, 880, {180, 216, 220, 264}}, {55, 888, {198, 207, 230, 253}},
{56, 912, {189, 216, 234, 273}},  {57, 925, {190, 228, 247, 260}},  {58, 1051, {145, 230, 299, 377}}, {59, 1386, {126, 198, 413, 649}}, {60, 969, {210, 234, 252, 273}},
{61, 1712, {126, 183, 549, 854}}, {62, 1003, {217, 234, 273, 279}}, {63, 1012, {231, 252, 253, 276}}, {64, 1044, {220, 240, 264, 320}}, {65, 1052, {234, 240, 272, 306}},
{66, 1067, {234, 252, 273, 308}}, {67, 1374, {168, 201, 469, 536}}, {68, 1107, {240, 255, 272, 340}}, {69, 1151, {208, 276, 299, 368}}, {70, 1125, {260, 273, 280, 312}},
{71, 1710, {171, 190, 639, 710}}, {72, 1155, {270, 280, 297, 308}}, {73, 1956, {204, 219, 292, 1241}},{74, 1205, {259, 280, 296, 370}},{75, 1217, {252, 300, 315, 350}},
{76, 1224, {272, 304, 306, 342}}, {77, 1247, {264, 308, 315, 360}}, {78, 1260, {273, 312, 315, 360}}, {79, 1602, {180, 316, 395, 711}}, {80, 1291, {285, 304, 342, 360}},
{81, 1300, {300, 324, 325, 351}}, {82, 1363, {275, 287, 350, 451}}, {83, 2036, {210, 249, 415, 1162}},{84, 1351, {312, 315, 360, 364}}, {85, 1368, {306, 340, 342, 380}},
{86, 1518, {253, 276, 473, 516}}, {87, 1430, {286, 319, 390, 435}}, {88, 1412, {330, 342, 360, 380}}, {89, 1990, {210, 267, 623, 890}}, {90, 1446, {336, 350, 360, 400}},
{91, 1487, {308, 330, 420, 429}}, {92, 1481, {342, 345, 380, 414}}, {93, 1549, {310, 340, 372, 527}}, {94, 1599, {282, 330, 470, 517}}, {95, 1538, {342, 345, 414, 437}},
{96, 1548, {352, 360, 396, 440}}, {97, 2568, {240, 291, 485,1552}},{98, 1583, {342, 380, 420, 441}}, {99, 1593, {352, 396, 416, 429}}, {100, 1615, {350, 390, 420, 455}},

1. Table[Module[{M = \[Infinity], S = {}, a, b, c, d, k, u, v}, For[a = 4 n, a > 2 n, a--, For[b = a + 1, b <= 4 n, b++, For[c = b + 1, c <= 9 n, c++, u = a*b*c*n;
   
2. v = a*b*c - n (a*b + (a + b) c); If[v > 0 && Mod[u, v] == 0, d = u/v;  If[c < d, k = a + b + c + d;  If[k < M, M = k; S = {a, b, c, d}]]]]]]; {n, M, S}], {n, 6, 100}]

复制代码

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作者: 王守恩    时间: 2026-1-15 06:32


楼上的代码搞错了1个符号。——2#的猜想很是有道理。

{{1,  24,  {2, 4, 6, 12}},     {2,  43,  {5,   6,  12, 20}},    {3,  52,  {9, 10, 15, 18}},        {4,  74,   {10,  15,  21,  28}},      {5,    84,   {15,  20,  21,  28}},
{6,  99,  {20, 21, 28, 30}}, {7, 120, {21, 24, 35, 40}},    {8, 135, {24, 30, 36, 45}},      {9, 153,  {26,  36,  39,  52}},     {10,  168,  {30,  40,  42,  56}},
{11, 180, {36, 44, 45, 55}}, {12, 198, {40, 42, 56, 60}},   {13, 216, {39, 52, 60, 65}},     {14, 229,  {45,  55,  63,  66}},    {15,  246,  {50,  55,  66,  75}},
{16, 261, {55, 60, 66, 80}},   {17, 308, {44, 68, 77, 119}},  {18, 297,  {60, 63, 84, 90}},     {19, 330,  {55,  66,  95, 114}},   {20,  323,  {70,  78,  84,  91}},
{21, 344, {70, 78, 91, 105}},  {22, 360,  {72,  88, 90, 110}},  {23, 405,  {60, 92, 115, 138}},  {24, 387,  {88,  90,   99, 110}},  {25,  417, {75, 100, 110, 132}},
{26, 420, {91, 104, 105, 120}}, {27, 440, {90, 108, 110, 132}}, {28, 458, {90, 110, 126, 132}}, {29, 546,  {78,  91, 174, 203}}, {30, 485, {108, 110, 132, 135}},
{31, 626, {72, 120, 155, 279}}, {32, 522, {110, 120, 132, 160}}, {33, 536, {112, 126, 144, 154}},{34, 583, {117, 119, 126, 221}},{35, 576, {112, 140, 144, 180}},
{36, 583, {126, 136, 153, 168}}, {37, 713, {84, 148, 222, 259}}, {38, 612, {136, 152, 153, 171}}, {39, 641, {136, 144, 153, 208}}, {40, 646, {140, 156, 168, 182}},
{41, 776, {120, 123, 205, 328}}, {42, 681, {147, 156, 182, 196}}, {43, 906, {129, 132, 172, 473}}, {44, 706, {165, 171, 180, 190}}, {45, 723, {168, 175, 180, 200}},
{46, 775, {156, 161, 182, 276}}, {47,1008, {112, 144, 329, 423}}, {48, 774, {176, 180, 198, 220}}, {49, 824, {147, 180, 245, 252}}, {50, 817, {175, 180, 210, 252}},
{51, 839, {176, 187, 204, 272}}, {52,  840, {182, 208, 210, 240}}, {53, 1189, {120, 168, 371, 530}}, {54, 880, {180, 216, 220, 264}}, {55, 888, {198, 207, 230, 253}},
{56, 912,  {189, 216, 234, 273}}, {57,  925, {190, 228, 247, 260}}, {58, 1051, {145, 230, 299, 377}}, {59, 1386, {126, 198, 413, 649}}, {60, 969, {210, 234, 252, 273}},
{61, 1464, {122, 244, 366, 732}}, {62, 1003, {217, 234, 273, 279}}, {63, 1012, {231, 252, 253, 276}}, {64, 1044, {220, 240, 264, 320}}, {65, 1052, {234, 240, 272, 306}},
{66, 1067, {234, 252, 273, 308}}, {67, 1374, {168, 201, 469, 536}}, {68, 1107, {240, 255, 272, 340}}, {69, 1151, {208, 276, 299, 368}}, {70, 1125, {260, 273, 280, 312}},
{71, 1704, {142, 284, 426, 852}}, {72, 1155, {270, 280, 297, 308}}, {73, 1752, {146, 292, 438, 876}}, {74, 1205, {259, 280, 296, 370}}, {75, 1217, {252, 300, 315, 350}},
{76, 1224, {272, 304, 306, 342}}, {77, 1247, {264, 308, 315, 360}}, {78, 1260, {273, 312, 315, 360}}, {79, 1602, {180, 316, 395, 711}}, {80, 1291, {285, 304, 342, 360}},
{81, 1300, {300, 324, 325, 351}}, {82, 1363, {275, 287, 350, 451}}, {83, 1992, {166, 332, 498, 996}}, {84, 1351, {312, 315, 360, 364}}, {85, 1368, {306, 340, 342, 380}},
{86, 1518, {253, 276, 473, 516}}, {87, 1430, {286, 319, 390, 435}}, {88, 1412, {330, 342, 360, 380}}, {89, 1990, {210, 267, 623, 890}}, {90, 1446, {336, 350, 360, 400}},
{91, 1487, {308, 330, 420, 429}}, {92, 1481, {342, 345, 380, 414}}, {93, 1549, {310, 340, 372, 527}}, {94, 1599, {282, 330, 470, 517}}, {95, 1538, {342, 345, 414, 437}},
{96, 1548, {352, 360, 396, 440}}, {97, 2328, {194, 388, 582,1164}},{98, 1583, {342, 380, 420, 441}}, {99, 1593, {352, 396, 416, 429}}, {100, 1615, {350, 390, 420, 455}},

1. Table[Module[{M = \[Infinity], S = {}, a, b, c, d, k, u, v}, For[a = 4 n, a ≥ 2 n, a--, For[b = a + 1, b <= 4 n, b++, For[c = b + 1, c <= 9 n, c++, u = a*b*c*n;
   
2. v = a*b*c - n (a*b + (a + b) c); If[v > 0 && Mod[u, v] == 0, d = u/v;  If[c < d, k = a + b + c + d;  If[k < M, M = k; S = {a, b, c, d}]]]]]]; {n, M, S}], {n, 100}]

复制代码

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作者: ysr    时间: 2026-1-16 17:30
https://www.toutiao.com/article/ ... /?wid=1768555423251

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作者: ysr    时间: 2026-1-16 17:35
1/7=1/8+1/56=1/9+1/72+1/56=1/10+1/90+1/72+1/56
10+90+72+56=228

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作者: ysr    时间: 2026-1-16 18:05
1/7=1/20+1/28+1/30+1/42=1/21+1/24+1/35+1/40
20+28+30+42=120
21+24+35+40=120

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作者: ysr    时间: 2026-1-16 18:13
这是另一种方法;
一般地，有如下方法将一个分数1/a拆成两个分数单位之和：

任选a的两个因数x和y；

将1/a的分子，分母同乘（x+y），得到x/a*(x+y)和y/a*(x+y)；

再将两个分数进行约分，得到两个分数单位之和。

扩展资料

若要将1/a拆成n个分数单位之和，可以任选a的n个因数，再按照上面的方法做。

把一个分数单位拆分成两个分数单位之和的方法是

⑴ 找分母的约数；

⑶ 拆分 把所得分数拆分成两个分数之和，使两个约数恰好是两个分数的分子；

⑷ 约分 把所得两个分数约成最简分数。

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作者: ysr    时间: 2026-1-16 18:17
网友点评：“仔细去看——2#,3#——有答案。7#的算法就是受2#,3#启发。”

答：谢谢！我好好学习一下！

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作者: ysr    时间: 2026-1-16 23:05


输入：7和130
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  124  {18,28,36,42}

输入：7和100
输出：无解

输入：7和300
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  179  {18,20,36,105}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  277  {14,20,63,180}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  198  {14,22,63,99}
7  244  {14,23,46,161}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  158  {14,27,54,63}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}

输入：7和700
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  361  {16,17,56,272}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  534  {15,22,35,462}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  438  {14,19,63,342}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  244  {14,23,46,161}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  424  {14,25,35,350}
7  158  {14,27,54,63}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}

输入：7和10000
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  1128  {19,21,24,1064}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}

代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(Text3)
a = Val(4 * n)

Do While a >= Val(2 * n) And a <= 4 * n
b = Val(a + 1)


Do While b <= Val(4 * n)
c = Val(b + 1)

Do While c <= Val(9 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = s2 & n & "  " & k & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a - 1)
Loop
If s3 > 0 Then
Text2 = s2
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-16 23:31
输入：7和120
输出：
7  120  {21,24,35,40}
7  120  {20,28,30,42}

输入：7和119
输出：无解

---

作者: ysr    时间: 2026-1-16 23:40
输入：7和20000
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  1128  {19,21,24,1064}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
解的个数为： 119组

显然有许多和值是相同的，和值有没有最大值以及解的组数是否是有限的，有待研究！本题很有趣！

又修改了一下，代码如下：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(Text3)
a = Val(4 * n)

Do While a >= Val(2 * n) And a <= 4 * n
b = Val(a + 1)


Do While b <= Val(4 * n)
c = Val(b + 1)

Do While c <= Val(9 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = s2 & n & "  " & k & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a - 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-16 23:45
输入：7和30000
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  1128  {19,21,24,1064}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
解的个数为： 120组

我对计算原理仍然不明白，代码是照着7#楼的那个代码猜测模仿而做的，谢谢网友！

---

作者: ysr    时间: 2026-1-16 23:46
输入：7和40000
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  1128  {19,21,24,1064}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
解的个数为： 120组

---

作者: ysr    时间: 2026-1-16 23:48
输入：7和50000
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  1128  {19,21,24,1064}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
解的个数为： 120组

---

作者: ysr    时间: 2026-1-16 23:51
输入：7和60000
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  1128  {19,21,24,1064}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
解的个数为： 120组

看来做多解的组数是：120组

---

作者: ysr    时间: 2026-1-16 23:55
和值不是按从小到大的顺序出现的，最大的和值好像是24428
（目测的，明天用程序比较大小找一下最大和值看看吧）

---

作者: ysr    时间: 2026-1-16 23:59
输入：7和612
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  361  {16,17,56,272}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  534  {15,22,35,462}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  438  {14,19,63,342}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  244  {14,23,46,161}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  424  {14,25,35,350}
7  158  {14,27,54,63}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
解的个数为： 79组

输入：7和168
输出：
7  129  {21,24,28,56}
7  120  {21,24,35,40}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  155  {15,24,56,60}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  158  {14,27,54,63}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
解的个数为： 19组

---

作者: ysr    时间: 2026-1-17 00:06
输入：6和70000
输出：
6  99  {20,21,28,30}
6  104  {18,20,30,36}
6  103  {18,21,28,36}
6  329  {16,17,24,272}
6  1063  {16,18,21,1008}
6  202  {16,18,24,144}
6  118  {16,18,36,48}
6  140  {16,20,24,80}
6  114  {16,20,30,48}
6  113  {16,21,28,48}
6  2218  {15,16,27,2160}
6  619  {15,16,28,560}
6  301  {15,16,30,240}
6  223  {15,16,32,160}
6  178  {15,16,35,112}
6  151  {15,16,40,80}
6  139  {15,16,48,60}
6  907  {15,17,25,850}
6  151  {15,17,34,85}
6  1091  {15,18,23,1035}
6  417  {15,18,24,360}
6  283  {15,18,25,225}
6  195  {15,18,27,135}
6  153  {15,18,30,90}
6  131  {15,18,35,63}
6  129  {15,18,36,60}
6  476  {15,20,21,420}
6  277  {15,20,22,220}
6  179  {15,20,24,120}
6  160  {15,20,25,100}
6  133  {15,20,28,70}
6  125  {15,20,30,60}
6  116  {15,20,36,45}
6  124  {15,21,28,60}
6  113  {15,21,35,42}
6  109  {15,24,30,40}
6  1325  {14,15,36,1260}
6  349  {14,15,40,280}
6  281  {14,15,42,210}
6  734  {14,16,32,672}
6  305  {14,16,35,240}
6  184  {14,16,42,112}
6  162  {14,16,48,84}
6  1487  {14,17,28,1428}
6  877  {14,18,26,819}
6  437  {14,18,27,378}
6  312  {14,18,28,252}
6  157  {14,18,35,90}
6  152  {14,18,36,84}
6  137  {14,18,42,63}
6  1121  {14,19,24,1064}
6  338  {14,20,24,280}
6  167  {14,20,28,105}
6  148  {14,20,30,84}
6  129  {14,20,35,60}
6  519  {14,21,22,462}
6  227  {14,21,24,168}
6  147  {14,21,28,84}
6  135  {14,21,30,70}
6  122  {14,24,28,56}
6  113  {14,24,35,40}
6  2932  {13,15,44,2860}
6  1243  {13,15,45,1170}
6  403  {13,15,50,325}
6  340  {13,15,52,260}
6  692  {13,16,39,624}
6  233  {13,16,48,156}
6  727  {13,17,34,663}
6  1231  {13,18,30,1170}
6  223  {13,18,36,156}
6  187  {13,18,39,117}
6  13397  {13,19,27,13338}
6  839  {13,20,26,780}
6  219  {13,20,30,156}
6  2242  {13,21,24,2184}
6  333  {13,21,26,273}
6  218  {13,21,28,156}
6  167  {13,24,26,104}
6  2429  {12,16,49,2352}
6  1278  {12,16,50,1200}
6  895  {12,16,51,816}
6  704  {12,16,52,624}
6  514  {12,16,54,432}
6  8434  {12,17,41,8364}
6  1499  {12,17,42,1428}
6  634  {12,17,44,561}
6  349  {12,17,48,272}
6  284  {12,17,51,204}
6  1399  {12,18,37,1332}
6  752  {12,18,38,684}
6  537  {12,18,39,468}
6  430  {12,18,40,360}
6  324  {12,18,42,252}
6  272  {12,18,44,198}
6  255  {12,18,45,180}
6  222  {12,18,48,144}
6  199  {12,18,52,117}
6  192  {12,18,54,108}
6  2572  {12,19,33,2508}
6  409  {12,19,36,342}
6  297  {12,19,38,228}
6  993  {12,20,31,930}
6  544  {12,20,32,480}
6  395  {12,20,33,330}
6  321  {12,20,34,255}
6  277  {12,20,35,210}
6  248  {12,20,36,180}
6  201  {12,20,39,130}
6  192  {12,20,40,120}
6  179  {12,20,42,105}
6  167  {12,20,45,90}
6  160  {12,20,48,80}
6  157  {12,20,50,75}
6  874  {12,21,29,812}
6  483  {12,21,30,420}
6  289  {12,21,32,224}
6  208  {12,21,35,140}
6  195  {12,21,36,126}
6  159  {12,21,42,84}
6  154  {12,21,44,77}
6  1249  {12,22,27,1188}
6  524  {12,22,28,462}
6  284  {12,22,30,220}
6  199  {12,22,33,132}
6  169  {12,22,36,99}
6  144  {12,22,44,66}
6  661  {12,24,25,600}
6  374  {12,24,26,312}
6  279  {12,24,27,216}
6  232  {12,24,28,168}
6  186  {12,24,30,120}
6  164  {12,24,32,96}
6  157  {12,24,33,88}
6  144  {12,24,36,72}
6  136  {12,24,40,60}
6  134  {12,24,42,56}
解的个数为： 136组

输入：6和100000
输出：
6  99  {20,21,28,30}
6  104  {18,20,30,36}
6  103  {18,21,28,36}
6  329  {16,17,24,272}
6  1063  {16,18,21,1008}
6  202  {16,18,24,144}
6  118  {16,18,36,48}
6  140  {16,20,24,80}
6  114  {16,20,30,48}
6  113  {16,21,28,48}
6  2218  {15,16,27,2160}
6  619  {15,16,28,560}
6  301  {15,16,30,240}
6  223  {15,16,32,160}
6  178  {15,16,35,112}
6  151  {15,16,40,80}
6  139  {15,16,48,60}
6  907  {15,17,25,850}
6  151  {15,17,34,85}
6  1091  {15,18,23,1035}
6  417  {15,18,24,360}
6  283  {15,18,25,225}
6  195  {15,18,27,135}
6  153  {15,18,30,90}
6  131  {15,18,35,63}
6  129  {15,18,36,60}
6  476  {15,20,21,420}
6  277  {15,20,22,220}
6  179  {15,20,24,120}
6  160  {15,20,25,100}
6  133  {15,20,28,70}
6  125  {15,20,30,60}
6  116  {15,20,36,45}
6  124  {15,21,28,60}
6  113  {15,21,35,42}
6  109  {15,24,30,40}
6  1325  {14,15,36,1260}
6  349  {14,15,40,280}
6  281  {14,15,42,210}
6  734  {14,16,32,672}
6  305  {14,16,35,240}
6  184  {14,16,42,112}
6  162  {14,16,48,84}
6  1487  {14,17,28,1428}
6  877  {14,18,26,819}
6  437  {14,18,27,378}
6  312  {14,18,28,252}
6  157  {14,18,35,90}
6  152  {14,18,36,84}
6  137  {14,18,42,63}
6  1121  {14,19,24,1064}
6  338  {14,20,24,280}
6  167  {14,20,28,105}
6  148  {14,20,30,84}
6  129  {14,20,35,60}
6  519  {14,21,22,462}
6  227  {14,21,24,168}
6  147  {14,21,28,84}
6  135  {14,21,30,70}
6  122  {14,24,28,56}
6  113  {14,24,35,40}
6  2932  {13,15,44,2860}
6  1243  {13,15,45,1170}
6  403  {13,15,50,325}
6  340  {13,15,52,260}
6  692  {13,16,39,624}
6  233  {13,16,48,156}
6  727  {13,17,34,663}
6  1231  {13,18,30,1170}
6  223  {13,18,36,156}
6  187  {13,18,39,117}
6  13397  {13,19,27,13338}
6  839  {13,20,26,780}
6  219  {13,20,30,156}
6  2242  {13,21,24,2184}
6  333  {13,21,26,273}
6  218  {13,21,28,156}
6  167  {13,24,26,104}
6  2429  {12,16,49,2352}
6  1278  {12,16,50,1200}
6  895  {12,16,51,816}
6  704  {12,16,52,624}
6  514  {12,16,54,432}
6  8434  {12,17,41,8364}
6  1499  {12,17,42,1428}
6  634  {12,17,44,561}
6  349  {12,17,48,272}
6  284  {12,17,51,204}
6  1399  {12,18,37,1332}
6  752  {12,18,38,684}
6  537  {12,18,39,468}
6  430  {12,18,40,360}
6  324  {12,18,42,252}
6  272  {12,18,44,198}
6  255  {12,18,45,180}
6  222  {12,18,48,144}
6  199  {12,18,52,117}
6  192  {12,18,54,108}
6  2572  {12,19,33,2508}
6  409  {12,19,36,342}
6  297  {12,19,38,228}
6  993  {12,20,31,930}
6  544  {12,20,32,480}
6  395  {12,20,33,330}
6  321  {12,20,34,255}
6  277  {12,20,35,210}
6  248  {12,20,36,180}
6  201  {12,20,39,130}
6  192  {12,20,40,120}
6  179  {12,20,42,105}
6  167  {12,20,45,90}
6  160  {12,20,48,80}
6  157  {12,20,50,75}
6  874  {12,21,29,812}
6  483  {12,21,30,420}
6  289  {12,21,32,224}
6  208  {12,21,35,140}
6  195  {12,21,36,126}
6  159  {12,21,42,84}
6  154  {12,21,44,77}
6  1249  {12,22,27,1188}
6  524  {12,22,28,462}
6  284  {12,22,30,220}
6  199  {12,22,33,132}
6  169  {12,22,36,99}
6  144  {12,22,44,66}
6  661  {12,24,25,600}
6  374  {12,24,26,312}
6  279  {12,24,27,216}
6  232  {12,24,28,168}
6  186  {12,24,30,120}
6  164  {12,24,32,96}
6  157  {12,24,33,88}
6  144  {12,24,36,72}
6  136  {12,24,40,60}
6  134  {12,24,42,56}
解的个数为： 136组

输入：6和99
输出：
6  99  {20,21,28,30}
解的个数为： 1组

输入：6和98
输出：无解

所以，99是1/6拆分成4个单位分数的和的分母和值的最小解。

1/6=1/20+1/21+1/28+1/30.

---

作者: ysr    时间: 2026-1-17 00:10
1/6拆分成4个单位分数的和的解的组数是：136组

可见，这类问题（拆分成4个单位分数之和）的解的组数可能是有限的。

---

作者: 王守恩    时间: 2026-1-17 04:39
代码A是7#的代码。代码B速度会快一些。\(M=\infty\ 改\ M=24\)——就是根据3#来的。又：网友指出{2,43,{5,6,12,20}}错了。正确——{2,41,{4,10,12,15}}。

1. 代码A——Table[Module[{M = \[Infinity], S = {}, a, b, c, d, k, u, v}, For[a = 4 n, a ≥ 2 n, a--, For[b = a + 1, b ≤ 4 n, b++, For[c = b + 1, c ≤ 9 n, c++, u = a*b*c*n;
   
2. v = a*b*c - n (a*b + (a + b) c); If[v > 0 && Mod[u, v] == 0, d = u/v;  If[c < d, k = a + b + c + d;  If[k < M, M = k; S = {a, b, c, d}]]]]]]; {n, M, S}], {n, 100}]

复制代码

1. 代码B——LaunchKernels[]; ParallelTable[Module[{M = 24 n, S = {}, a, b, c, d, k, u, v}, For[a = 2 n, a < 4 n, a++, For[b = a + 1, b ≤ 4 n, b++, For[c = b + 1, c ≤ 9 n, c++, u = a*b*c*n;
   
2. v = a*b*c - n (a*b + (a + b) c); If[v > 0 && Mod[u, v] == 0, d = u/v; If[c < d && d ≤ 12 n, k = a + b + c + d; If[k ≤ M, M = k; S = {a, b, c, d}]]]]]]; {n, M, S}], {n, 100}]

复制代码

{{1,  24,  {2,   4,   6, 12}},  {2,   43,  {5,   6,   12,  20}}, {3,  52,  {9,   10,  15, 18}},    {4,  74,   {10,  15,   21,  28}},   {5,     84,    {15,  20,   21,  28}},
{6,  99,  {20, 21, 28, 30}},  {7,  120, {21,  24,  35,  40}}, {8, 135, {24,  30,  36, 45}},    {9,  153,  {26,  36,   39,  52}},   {10,  168,   {30,  40,   42,  56}},
{11, 180, {36, 44, 45, 55}}, {12, 198, {40,   42,  56,  60}}, {13, 216, {39,  52,  60, 65}},   {14,  229,  {45,  55,   63,  66}},   {15,  246,   {50,  55,   66,  75}},
{16, 261, {55, 60,  66,  80}}, {17, 308, {44,   68,  77, 119}}, {18, 297, {60,  63,  84, 90}},   {19,  330,  {55,  66,   95, 114}},   {20, 323,   {70,  78,   84,  91}},
{21, 344, {70,  78,  91, 105}}, {22, 360, {72,   88,  90, 110}}, {23, 405, {60,  92,  115, 138}}, {24,  387,  {88,  90,   99, 110}},   {25, 417,  {75,  100, 110, 132}},
{26, 420, {91, 104, 105, 120}}, {27, 440, {90, 108, 110, 132}}, {28, 458, {90,  110,  126, 132}}, {29, 546,  {78,  91,  174, 203}}, {30, 485,  {108, 110, 132, 135}},
{31, 626, {72, 120, 155, 279}}, {32, 522, {110, 120, 132, 160}}, {33, 536, {112, 126, 144, 154}}, {34, 583, {117, 119, 126, 221}}, {35, 576,  {112, 140, 144, 180}},
{36, 583, {126, 136, 153, 168}}, {37, 713, {84,  148, 222, 259}}, {38, 612, {136, 152, 153, 171}}, {39, 641, {136, 144, 153, 208}}, {40, 646,  {140, 156, 168, 182}},
{41, 776, {120, 123, 205, 328}}, {42, 681, {147, 156, 182, 196}}, {43, 906, {129, 132, 172, 473}}, {44, 706, {165, 171, 180, 190}}, {45, 723,  {168, 175, 180, 200}},
{46, 775, {156, 161, 182, 276}}, {47, 1008, {112, 144, 329, 423}},{48,  774, {176, 180, 198, 220}}, {49, 824, {147, 180, 245, 252}}, {50, 817,  {175, 180, 210, 252}},
{51, 839, {176, 187, 204, 272}}, {52,  840,  {182, 208, 210, 240}}, {53, 1189, {120, 168, 371, 530}}, {54, 880, {180, 216, 220, 264}}, {55, 888,  {198, 207, 230, 253}},
{56, 912, {189, 216, 234, 273}}, {57,  925,  {190, 228, 247, 260}}, {58, 1051, {145, 230, 299, 377}}, {59, 1386, {126, 198, 413, 649}}, {60, 969,  {210, 234, 252, 273}},
{61, 1464, {122, 244, 366, 732}}, {62, 1003, {217, 234, 273, 279}}, {63, 1012, {231, 252, 253, 276}}, {64, 1044, {220, 240, 264, 320}}, {65, 1052, {234, 240, 272, 306}},
{66, 1067, {234, 252, 273, 308}}, {67, 1374, {168, 201, 469, 536}}, {68, 1107, {240, 255, 272, 340}}, {69, 1151, {208, 276, 299, 368}}, {70, 1125, {260, 273, 280, 312}},
{71, 1704, {142, 284, 426, 852}}, {72, 1155, {270, 280, 297, 308}}, {73, 1752, {146, 292, 438, 876}}, {74, 1205, {259, 280, 296, 370}}, {75, 1217, {252, 300, 315, 350}},
{76, 1224, {272, 304, 306, 342}}, {77, 1247, {264, 308, 315, 360}}, {78, 1260, {273, 312, 315, 360}}, {79, 1602, {180, 316, 395, 711}}, {80, 1291, {285, 304, 342, 360}},
{81, 1300, {300, 324, 325, 351}}, {82, 1363, {275, 287, 350, 451}}, {83, 1992, {166, 332, 498, 996}}, {84, 1351, {312, 315, 360, 364}}, {85, 1368, {306, 340, 342, 380}},
{86, 1518, {253, 276, 473, 516}}, {87, 1430, {286, 319, 390, 435}}, {88, 1412, {330, 342, 360, 380}}, {89, 1990, {210, 267, 623, 890}}, {90, 1446, {336, 350, 360, 400}},
{91, 1487, {308, 330, 420, 429}}, {92, 1481, {342, 345, 380, 414}}, {93, 1549, {310, 340, 372, 527}}, {94, 1599, {282, 330, 470, 517}}, {95, 1538, {342, 360, 380, 456}},
{96, 1548, {352, 360, 396, 440}}, {97, 2328, {194, 388, 582,1164}}, {98, 1583, {342, 380, 420, 441}}, {99,1593, {352, 396, 416, 429}}, {100, 1615, {350, 390, 420, 455}}}

---

作者: ysr    时间: 2026-1-17 07:06
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
7  184  {14,30,35,105}
7  156  {14,30,42,70}
7  152  {14,30,45,63}
7  145  {14,35,40,56}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  150  {15,30,35,70}
7  141  {15,30,40,56}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1128  {19,21,24,1064}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  129  {21,24,28,56}
7  120  {21,24,35,40}
解的个数为： 126组

---

作者: ysr    时间: 2026-1-17 08:02


7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
7  184  {14,30,35,105}
7  156  {14,30,42,70}
7  152  {14,30,45,63}
7  145  {14,35,40,56}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  150  {15,30,35,70}
7  141  {15,30,40,56}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1128  {19,21,24,1064}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  129  {21,24,28,56}
7  120  {21,24,35,40}
解的个数为： 126组

修改后代码如下：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(Text3)
a = Val(2 * n)

Do While a >= Val(2 * n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(5 * n)
c = Val(b + 1)

Do While c <= Val(9 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = s2 & n & "  " & k & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 08:11
再次修改后的结果：
输入：7和50000
输出：
7  9631  {14,17,80,9520}
7  1543  {14,17,84,1428}
7  1306  {14,17,85,1190}
7  490  {14,17,102,357}
7  415  {14,17,112,272}
7  4128  {14,18,64,4032}
7  1484  {14,18,66,1386}
7  732  {14,18,70,630}
7  608  {14,18,72,504}
7  368  {14,18,84,252}
7  332  {14,18,90,210}
7  288  {14,18,112,144}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  244  {14,20,70,140}
7  226  {14,20,80,112}
7  223  {14,20,84,105}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  210  {14,21,70,105}
7  204  {14,21,78,91}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
7  184  {14,30,35,105}
7  156  {14,30,42,70}
7  152  {14,30,45,63}
7  145  {14,35,40,56}
7  2906  {15,16,75,2800}
7  951  {15,16,80,840}
7  675  {15,16,84,560}
7  376  {15,16,105,240}
7  353  {15,16,112,210}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  228  {15,18,90,105}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  189  {15,20,70,84}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  150  {15,30,35,70}
7  141  {15,30,40,56}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1128  {19,21,24,1064}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  129  {21,24,28,56}
7  120  {21,24,35,40}
解的个数为： 150组

---

作者: ysr    时间: 2026-1-17 08:15
7  44550  {14,15,211,44310}
7  22501  {14,15,212,22260}
7  15152  {14,15,213,14910}
7  11478  {14,15,214,11235}
7  9274  {14,15,215,9030}
7  7805  {14,15,216,7560}
7  6756  {14,15,217,6510}
7  5358  {14,15,219,5110}
7  4869  {14,15,220,4620}
7  4136  {14,15,222,3885}
7  3613  {14,15,224,3360}
7  3404  {14,15,225,3150}
7  2917  {14,15,228,2660}
7  2674  {14,15,230,2415}
7  2570  {14,15,231,2310}
7  2238  {14,15,235,1974}
7  2052  {14,15,238,1785}
7  1949  {14,15,240,1680}
7  1744  {14,15,245,1470}
7  1710  {14,15,246,1435}
7  1541  {14,15,252,1260}
7  1474  {14,15,255,1190}
7  1398  {14,15,259,1110}
7  1381  {14,15,260,1092}
7  1244  {14,15,270,945}
7  1212  {14,15,273,910}
7  1149  {14,15,280,840}
7  1112  {14,15,285,798}
7  1058  {14,15,294,735}
7  1029  {14,15,300,700}
7  997  {14,15,308,660}
7  990  {14,15,310,651}
7  974  {14,15,315,630}
7  925  {14,15,336,560}
7  904  {14,15,350,525}
7  12799  {14,16,113,12656}
7  6528  {14,16,114,6384}
7  3394  {14,16,116,3248}
7  2053  {14,16,119,1904}
7  1830  {14,16,120,1680}
7  1164  {14,16,126,1008}
7  1054  {14,16,128,896}
7  730  {14,16,140,560}
7  678  {14,16,144,504}
7  559  {14,16,161,368}
7  534  {14,16,168,336}
7  514  {14,16,176,308}
7  480  {14,16,210,240}
7  9631  {14,17,80,9520}
7  1543  {14,17,84,1428}
7  1306  {14,17,85,1190}
7  490  {14,17,102,357}
7  415  {14,17,112,272}
7  388  {14,17,119,238}
7  4128  {14,18,64,4032}
7  1484  {14,18,66,1386}
7  732  {14,18,70,630}
7  608  {14,18,72,504}
7  368  {14,18,84,252}
7  332  {14,18,90,210}
7  288  {14,18,112,144}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  244  {14,20,70,140}
7  226  {14,20,80,112}
7  223  {14,20,84,105}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  210  {14,21,70,105}
7  204  {14,21,78,91}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
7  184  {14,30,35,105}
7  156  {14,30,42,70}
7  152  {14,30,45,63}
7  145  {14,35,40,56}
7  2906  {15,16,75,2800}
7  951  {15,16,80,840}
7  675  {15,16,84,560}
7  376  {15,16,105,240}
7  353  {15,16,112,210}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  228  {15,18,90,105}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  189  {15,20,70,84}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  150  {15,30,35,70}
7  141  {15,30,40,56}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1128  {19,21,24,1064}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  129  {21,24,28,56}
7  120  {21,24,35,40}
解的个数为： 199组

再次修改后的代码如下：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(Text3)
a = Val(2 * n)

Do While a >= Val(2 * n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(5 * n)
c = Val(b + 1)

Do While c <= Val(50 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = s2 & n & "  " & k & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 08:24
再次修改后的程序结果：
输入：7和50000
输出：
7  40753  {8,67,344,40334}
7  13825  {8,67,350,13400}
7  38476  {8,68,320,38080}
7  22294  {8,68,322,21896}
7  18487  {8,68,323,18088}
7  10162  {8,68,328,9758}
7  6124  {8,68,336,5712}
7  5176  {8,68,340,4760}
7  3826  {8,68,350,3400}
7  32577  {8,69,300,32200}
7  24114  {8,69,301,23736}
7  6668  {8,69,312,6279}
7  4263  {8,69,322,3864}
7  2989  {8,69,336,2576}
7  39840  {8,70,282,39480}
7  20242  {8,70,284,19880}
7  16323  {8,70,285,15960}
7  11845  {8,70,287,11480}
7  10446  {8,70,288,10080}
7  8488  {8,70,290,8120}
7  6252  {8,70,294,5880}
7  5554  {8,70,296,5180}
7  4578  {8,70,300,4200}
7  3799  {8,70,305,3416}
7  3466  {8,70,308,3080}
7  3120  {8,70,312,2730}
7  2913  {8,70,315,2520}
7  2638  {8,70,320,2240}
7  2287  {8,70,329,1880}
7  2256  {8,70,330,1848}
7  2094  {8,70,336,1680}
7  1927  {8,70,344,1505}
7  1828  {8,70,350,1400}
7  5329  {8,71,280,4970}
7  4339  {8,71,284,3976}
7  32338  {8,72,254,32004}
7  21755  {8,72,255,21420}
7  16464  {8,72,256,16128}
7  11174  {8,72,258,10836}
7  9663  {8,72,259,9324}
7  8530  {8,72,260,8190}
7  7649  {8,72,261,7308}
7  5888  {8,72,264,5544}
7  5134  {8,72,266,4788}
7  4569  {8,72,268,4221}
7  4130  {8,72,270,3780}
7  3629  {8,72,273,3276}
7  3254  {8,72,276,2898}
7  2963  {8,72,279,2604}
7  2880  {8,72,280,2520}
7  2384  {8,72,288,2016}
7  2138  {8,72,294,1764}
7  1955  {8,72,300,1575}
7  1929  {8,72,301,1548}
7  1814  {8,72,306,1428}
7  1774  {8,72,308,1386}
7  1655  {8,72,315,1260}
7  1538  {8,72,324,1134}
7  1449  {8,72,333,1036}
7  1424  {8,72,336,1008}
7  1330  {8,72,350,900}
7  5589  {8,73,252,5256}
7  30358  {8,74,232,30044}
7  2998  {8,74,252,2664}
7  2413  {8,74,259,2072}
7  1657  {8,74,280,1295}
7  1414  {8,74,296,1036}
7  17107  {8,75,224,16800}
7  12908  {8,75,225,12600}
7  3123  {8,75,240,2800}
7  2135  {8,75,252,1800}
7  1413  {8,75,280,1050}
7  1223  {8,75,300,840}
7  1033  {8,75,350,600}
7  14664  {8,76,216,14364}
7  4564  {8,76,224,4256}
7  3504  {8,76,228,3192}
7  1704  {8,76,252,1368}
7  1414  {8,76,266,1064}
7  1209  {8,76,285,840}
7  16309  {8,77,208,16016}
7  11998  {8,77,209,11704}
7  9535  {8,77,210,9240}
7  4459  {8,77,216,4158}
7  3385  {8,77,220,3080}
7  2773  {8,77,224,2464}
7  2164  {8,77,231,1848}
7  1819  {8,77,238,1496}
7  1273  {8,77,264,924}
7  1135  {8,77,280,770}
7  1099  {8,77,286,728}
7  1009  {8,77,308,616}
7  949  {8,77,336,528}
7  27586  {8,78,200,27300}
7  9337  {8,78,203,9048}
7  4662  {8,78,208,4368}
7  3936  {8,78,210,3640}
7  2759  {8,78,216,2457}
7  1274  {8,78,252,936}
7  1186  {8,78,260,840}
7  1087  {8,78,273,728}
7  944  {8,78,312,546}
7  26596  {8,80,188,26320}
7  15397  {8,80,189,15120}
7  10918  {8,80,190,10640}
7  7000  {8,80,192,6720}
7  4651  {8,80,195,4368}
7  4204  {8,80,196,3920}
7  3088  {8,80,200,2800}
7  2611  {8,80,203,2320}
7  2116  {8,80,208,1820}
7  1978  {8,80,210,1680}
7  1540  {8,80,220,1232}
7  1432  {8,80,224,1120}
7  1168  {8,80,240,840}
7  1117  {8,80,245,784}
7  1060  {8,80,252,720}
7  955  {8,80,272,595}
7  928  {8,80,280,560}
7  856  {8,80,320,448}
7  844  {8,80,336,420}
7  838  {8,80,350,400}
7  13314  {8,81,184,13041}
7  4814  {8,81,189,4536}
7  1439  {8,81,216,1134}
7  989  {8,81,252,648}
7  28653  {8,84,169,28392}
7  14542  {8,84,170,14280}
7  9839  {8,84,171,9576}
7  7488  {8,84,172,7224}
7  5138  {8,84,174,4872}
7  4467  {8,84,175,4200}
7  3964  {8,84,176,3696}
7  3573  {8,84,177,3304}
7  2792  {8,84,180,2520}
7  2458  {8,84,182,2184}
7  2208  {8,84,184,1932}
7  2014  {8,84,186,1736}
7  1793  {8,84,189,1512}
7  1628  {8,84,192,1344}
7  1464  {8,84,196,1176}
7  1342  {8,84,200,1050}
7  1248  {8,84,204,952}
7  1142  {8,84,210,840}
7  1064  {8,84,216,756}
7  1053  {8,84,217,744}
7  988  {8,84,224,672}
7  939  {8,84,231,616}
7  933  {8,84,232,609}
7  892  {8,84,240,560}
7  848  {8,84,252,504}
7  818  {8,84,264,462}
7  814  {8,84,266,456}
7  792  {8,84,280,420}
7  778  {8,84,294,392}
7  768  {8,84,312,364}
7  767  {8,84,315,360}
7  31674  {8,85,165,31416}
7  7401  {8,85,168,7140}
7  5023  {8,85,170,4760}
7  1137  {8,85,204,840}
7  3874  {8,86,168,3612}
7  2674  {8,86,172,2408}
7  739  {8,86,301,344}
7  2699  {8,87,168,2436}
7  1893  {8,87,174,1624}
7  994  {8,87,203,696}
7  24121  {8,88,155,23870}
7  12264  {8,88,156,12012}
7  6337  {8,88,158,6083}
7  3799  {8,88,161,3542}
7  2571  {8,88,165,2310}
7  2112  {8,88,168,1848}
7  1504  {8,88,176,1232}
7  1279  {8,88,182,1001}
7  987  {8,88,198,693}
7  937  {8,88,203,638}
7  789  {8,88,231,462}
7  744  {8,88,252,396}
7  721  {8,88,275,350}
7  23923  {8,89,152,23674}
7  8083  {8,89,154,7832}
7  12848  {8,90,150,12600}
7  6235  {8,90,152,5985}
7  5011  {8,90,153,4760}
7  4212  {8,90,154,3960}
7  2274  {8,90,160,2016}
7  1526  {8,90,168,1260}
7  1118  {8,90,180,840}
7  812  {8,90,210,504}
7  710  {8,90,252,360}
7  693  {8,90,280,315}
7  15534  {8,91,147,15288}
7  3709  {8,91,152,3458}
7  2439  {8,91,156,2184}
7  1359  {8,91,168,1092}
7  1009  {8,91,182,728}
7  739  {8,91,224,416}
7  684  {8,91,273,312}
7  23428  {8,92,144,23184}
7  2278  {8,92,154,2024}
7  1549  {8,92,161,1288}
7  1234  {8,92,168,966}
7  928  {8,92,184,644}
7  1137  {8,93,168,868}
7  13402  {8,94,140,13160}
7  8139  {8,94,141,7896}
7  5563  {8,95,140,5320}
7  1585  {8,95,152,1330}
7  649  {8,95,266,280}
7  30479  {8,96,135,30240}
7  11664  {8,96,136,11424}
7  5394  {8,96,138,5152}
7  3604  {8,96,140,3360}
7  2264  {8,96,144,2016}
7  1819  {8,96,147,1568}
7  1314  {8,96,154,1056}
7  1104  {8,96,160,840}
7  944  {8,96,168,672}
7  744  {8,96,192,448}
7  664  {8,96,224,336}
7  644  {8,96,252,288}
7  13174  {8,98,132,12936}
7  7687  {8,98,133,7448}
7  3574  {8,98,136,3332}
7  2206  {8,98,140,1960}
7  1429  {8,98,147,1176}
7  1189  {8,98,152,931}
7  862  {8,98,168,588}
7  694  {8,98,196,392}
7  631  {8,98,245,280}
7  5783  {8,99,132,5544}
7  1483  {8,99,144,1232}
7  1053  {8,99,154,792}
7  623  {8,99,252,264}
7  22636  {8,100,128,22400}
7  1648  {8,100,140,1400}
7  1098  {8,100,150,840}
7  801  {8,100,168,525}
7  658  {8,100,200,350}
7  8804  {8,102,126,8568}
7  1674  {8,102,136,1428}
7  754  {8,102,168,476}
7  22438  {8,104,122,22204}
7  5878  {8,104,124,5642}
7  3514  {8,104,126,3276}
7  2062  {8,104,130,1820}
7  1162  {8,104,140,910}
7  838  {8,104,154,572}
7  814  {8,104,156,546}
7  658  {8,104,182,364}
7  598  {8,104,234,252}
7  14754  {8,105,121,14520}
7  7555  {8,105,122,7320}
7  5156  {8,105,123,4920}
7  3957  {8,105,124,3720}
7  3238  {8,105,125,3000}
7  2759  {8,105,126,2520}
7  2161  {8,105,128,1920}
7  1962  {8,105,129,1720}
7  1803  {8,105,130,1560}
7  1565  {8,105,132,1320}
7  1328  {8,105,135,1080}
7  1269  {8,105,136,1020}
7  1171  {8,105,138,920}
7  1093  {8,105,140,840}
7  977  {8,105,144,720}
7  954  {8,105,145,696}
7  863  {8,105,150,600}
7  835  {8,105,152,570}
7  789  {8,105,156,520}
7  753  {8,105,160,480}
7  718  {8,105,165,440}
7  701  {8,105,168,420}
7  691  {8,105,170,408}
7  653  {8,105,180,360}
7  642  {8,105,184,345}
7  625  {8,105,192,320}
7  620  {8,105,195,312}
7  613  {8,105,200,300}
7  603  {8,105,210,280}
7  599  {8,105,216,270}
7  597  {8,105,220,264}
7  11364  {8,106,120,11130}
7  19889  {8,108,117,19656}
7  4016  {8,108,120,3780}
7  1754  {8,108,126,1512}
7  1091  {8,108,135,840}
7  662  {8,108,168,378}
7  584  {8,108,216,252}
7  14401  {8,110,115,14168}
7  2548  {8,110,120,2310}
7  1090  {8,110,132,840}
7  874  {8,110,140,616}
7  712  {8,110,154,440}
7  12889  {8,112,113,12656}
7  6618  {8,112,114,6384}
7  3484  {8,112,116,3248}
7  2143  {8,112,119,1904}
7  1920  {8,112,120,1680}
7  1254  {8,112,126,1008}
7  1144  {8,112,128,896}
7  820  {8,112,140,560}
7  768  {8,112,144,504}
7  649  {8,112,161,368}
7  624  {8,112,168,336}
7  604  {8,112,176,308}
7  570  {8,112,210,240}
7  1572  {8,114,120,1330}
7  673  {8,114,152,399}
7  559  {8,116,203,232}
7  979  {8,117,126,728}
7  739  {8,119,136,476}
7  577  {8,119,170,280}
7  884  {8,120,126,630}
7  804  {8,120,130,546}
7  688  {8,120,140,420}
7  628  {8,120,150,350}
7  612  {8,120,154,330}
7  576  {8,120,168,280}
7  560  {8,120,180,252}
7  634  {8,126,140,360}
7  614  {8,126,144,336}
7  554  {8,126,168,252}
7  539  {8,126,189,216}
7  33752  {9,35,318,33390}
7  20524  {9,35,320,20160}
7  14856  {9,35,322,14490}
7  11708  {9,35,324,11340}
7  7304  {9,35,330,6930}
7  5420  {9,35,336,5040}
7  4668  {9,35,340,4284}
7  4376  {9,35,342,3990}
7  3544  {9,35,350,3150}
7  32303  {9,36,254,32004}
7  21720  {9,36,255,21420}
7  16429  {9,36,256,16128}
7  11139  {9,36,258,10836}
7  9628  {9,36,259,9324}
7  8495  {9,36,260,8190}
7  7614  {9,36,261,7308}
7  5853  {9,36,264,5544}
7  5099  {9,36,266,4788}
7  4534  {9,36,268,4221}
7  4095  {9,36,270,3780}
7  3594  {9,36,273,3276}
7  3219  {9,36,276,2898}
7  2928  {9,36,279,2604}
7  2845  {9,36,280,2520}
7  2349  {9,36,288,2016}
7  2103  {9,36,294,1764}
7  1920  {9,36,300,1575}
7  1894  {9,36,301,1548}
7  1779  {9,36,306,1428}
7  1739  {9,36,308,1386}
7  1620  {9,36,315,1260}
7  1503  {9,36,324,1134}
7  1414  {9,36,333,1036}
7  1389  {9,36,336,1008}
7  1295  {9,36,350,900}
7  4930  {9,37,222,4662}
7  1630  {9,37,252,1332}
7  7418  {9,38,189,7182}
7  6222  {9,38,190,5985}
7  983  {9,38,252,684}
7  788  {9,38,342,399}
7  6768  {9,39,168,6552}
7  2048  {9,39,180,1820}
7  1868  {9,39,182,1638}
7  828  {9,39,234,546}
7  768  {9,39,252,468}
7  12799  {9,40,150,12600}
7  6186  {9,40,152,5985}
7  4962  {9,40,153,4760}
7  4163  {9,40,154,3960}
7  2225  {9,40,160,2016}
7  1477  {9,40,168,1260}
7  1069  {9,40,180,840}
7  763  {9,40,210,504}
7  661  {9,40,252,360}
7  644  {9,40,280,315}
7  16180  {9,42,127,16002}
7  8243  {9,42,128,8064}
7  5598  {9,42,129,5418}
7  4276  {9,42,130,4095}
7  2955  {9,42,132,2772}
7  2578  {9,42,133,2394}
7  2076  {9,42,135,1890}
7  1638  {9,42,138,1449}
7  1451  {9,42,140,1260}
7  1203  {9,42,144,1008}
7  1080  {9,42,147,882}
7  918  {9,42,153,714}
7  898  {9,42,154,693}
7  780  {9,42,162,567}
7  723  {9,42,168,504}
7  676  {9,42,175,450}
7  651  {9,42,180,420}
7  618  {9,42,189,378}
7  580  {9,42,207,322}
7  576  {9,42,210,315}
7  563  {9,42,224,288}
7  558  {9,42,234,273}
7  1984  {9,43,126,1806}
7  11253  {9,44,112,11088}
7  1103  {9,44,126,924}
7  878  {9,44,132,693}
7  603  {9,44,154,396}
7  503  {9,44,198,252}
7  11290  {9,45,106,11130}
7  3942  {9,45,108,3780}
7  2474  {9,45,110,2310}
7  1846  {9,45,112,1680}
7  1498  {9,45,114,1330}
7  1014  {9,45,120,840}
7  810  {9,45,126,630}

---

作者: ysr    时间: 2026-1-17 08:24
7  730  {9,45,130,546}
7  614  {9,45,140,420}
7  554  {9,45,150,350}
7  538  {9,45,154,330}
7  502  {9,45,168,280}
7  486  {9,45,180,252}
7  2230  {9,46,105,2070}
7  664  {9,46,126,483}
7  23333  {9,48,92,23184}
7  2169  {9,48,96,2016}
7  1388  {9,48,99,1232}
7  882  {9,48,105,720}
7  673  {9,48,112,504}
7  519  {9,48,126,336}
7  453  {9,48,144,252}
7  4558  {9,49,90,4410}
7  1038  {9,49,98,882}
7  478  {9,49,126,294}
7  1724  {9,50,90,1575}
7  614  {9,50,105,450}
7  4428  {9,51,84,4284}
7  424  {9,51,126,238}
7  1783  {9,52,84,1638}
7  430  {9,52,117,252}
7  14503  {9,54,76,14364}
7  4298  {9,54,77,4158}
7  2598  {9,54,78,2457}
7  1278  {9,54,81,1134}
7  903  {9,54,84,756}
7  438  {9,54,105,270}
7  423  {9,54,108,252}
7  378  {9,54,126,189}
7  5394  {9,56,73,5256}
7  2803  {9,56,74,2664}
7  1940  {9,56,75,1800}
7  1509  {9,56,76,1368}
7  1079  {9,56,78,936}
7  865  {9,56,80,720}
7  794  {9,56,81,648}
7  653  {9,56,84,504}
7  549  {9,56,88,396}
7  515  {9,56,90,360}
7  449  {9,56,96,288}
7  428  {9,56,99,264}
7  403  {9,56,104,234}
7  389  {9,56,108,216}
7  365  {9,56,120,180}
7  359  {9,56,126,168}
7  354  {9,56,136,153}
7  3330  {9,57,72,3192}
7  1763  {9,58,72,1624}
7  1399  {9,60,70,1260}
7  981  {9,60,72,840}
7  468  {9,60,84,315}
7  411  {9,60,90,252}
7  354  {9,60,105,180}
7  335  {9,60,126,140}
7  4168  {9,63,64,4032}
7  1524  {9,63,66,1386}
7  772  {9,63,70,630}
7  648  {9,63,72,504}
7  408  {9,63,84,252}
7  372  {9,63,90,210}
7  328  {9,63,112,144}
7  593  {9,64,72,448}
7  328  {9,66,99,154}
7  431  {9,70,72,280}
7  343  {9,70,84,180}
7  310  {9,70,105,126}
7  333  {9,72,84,168}
7  323  {9,72,88,154}
7  306  {9,72,105,120}
7  304  {9,78,91,126}
7  21196  {10,26,230,20930}
7  15282  {10,26,231,15015}
7  8460  {10,26,234,8190}
7  4644  {10,26,240,4368}
7  3466  {10,26,245,3185}
7  2628  {10,26,252,2340}
7  2116  {10,26,260,1820}
7  1674  {10,26,273,1365}
7  1188  {10,26,312,840}
7  1170  {10,26,315,819}
7  1036  {10,26,350,650}
7  9662  {10,27,175,9450}
7  3997  {10,27,180,3780}
7  2116  {10,27,189,1890}
7  1192  {10,27,210,945}
7  1093  {10,27,216,840}
7  829  {10,27,252,540}
7  730  {10,27,315,378}
7  19919  {10,28,141,19740}
7  10120  {10,28,142,9940}
7  5222  {10,28,144,5040}
7  4243  {10,28,145,4060}
7  3125  {10,28,147,2940}
7  2776  {10,28,148,2590}
7  2288  {10,28,150,2100}
7  1732  {10,28,154,1540}
7  1559  {10,28,156,1365}
7  1318  {10,28,160,1120}
7  1127  {10,28,165,924}
7  1046  {10,28,168,840}
7  913  {10,28,175,700}
7  848  {10,28,180,630}
7  767  {10,28,189,540}
7  760  {10,28,190,532}
7  724  {10,28,196,490}
7  668  {10,28,210,420}
7  643  {10,28,220,385}
7  616  {10,28,238,340}
7  614  {10,28,240,336}
7  605  {10,28,252,315}
7  24519  {10,29,120,24360}
7  991  {10,29,140,812}
7  532  {10,29,203,290}
7  11276  {10,30,106,11130}
7  3928  {10,30,108,3780}
7  2460  {10,30,110,2310}
7  1832  {10,30,112,1680}
7  1484  {10,30,114,1330}
7  1000  {10,30,120,840}
7  796  {10,30,126,630}
7  716  {10,30,130,546}
7  600  {10,30,140,420}
7  540  {10,30,150,350}
7  524  {10,30,154,330}
7  488  {10,30,168,280}
7  472  {10,30,180,252}
7  1076  {10,31,105,930}
7  2148  {10,32,90,2016}
7  978  {10,32,96,840}
7  627  {10,32,105,480}
7  406  {10,32,140,224}
7  18603  {10,33,80,18480}
7  1667  {10,33,84,1540}
7  971  {10,33,88,840}
7  826  {10,33,90,693}
7  478  {10,33,105,330}
7  362  {10,33,154,165}
7  9044  {10,34,75,8925}
7  724  {10,34,85,595}
7  404  {10,34,105,255}
7  5086  {10,35,71,4970}
7  2637  {10,35,72,2520}
7  1414  {10,35,74,1295}
7  1170  {10,35,75,1050}
7  892  {10,35,77,770}
7  685  {10,35,80,560}
7  549  {10,35,84,420}
7  450  {10,35,90,315}
7  406  {10,35,95,266}
7  388  {10,35,98,245}
7  360  {10,35,105,210}
7  334  {10,35,119,170}
7  333  {10,35,120,168}
7  1376  {10,36,70,1260}
7  958  {10,36,72,840}
7  445  {10,36,84,315}
7  388  {10,36,90,252}
7  331  {10,36,105,180}
7  312  {10,36,126,140}
7  1929  {10,39,60,1820}
7  660  {10,39,65,546}
7  284  {10,39,105,130}
7  3299  {10,40,57,3192}
7  1732  {10,40,58,1624}
7  950  {10,40,60,840}
7  617  {10,40,63,504}
7  562  {10,40,64,448}
7  400  {10,40,70,280}
7  374  {10,40,72,252}
7  302  {10,40,84,168}
7  292  {10,40,88,154}
7  275  {10,40,105,120}
7  1747  {10,41,56,1640}
7  5670  {10,42,53,5565}
7  1996  {10,42,54,1890}
7  1262  {10,42,55,1155}
7  948  {10,42,56,840}
7  774  {10,42,57,665}
7  532  {10,42,60,420}
7  430  {10,42,63,315}
7  390  {10,42,65,273}
7  332  {10,42,70,210}
7  302  {10,42,75,175}
7  294  {10,42,77,165}
7  276  {10,42,84,140}
7  268  {10,42,90,126}
7  7804  {10,44,50,7700}
7  550  {10,44,56,440}
7  271  {10,44,77,140}
7  4514  {10,45,49,4410}
7  1680  {10,45,50,1575}
7  471  {10,45,56,360}
7  367  {10,45,60,252}
7  328  {10,45,63,210}
7  250  {10,45,90,105}
7  354  {10,48,56,240}
7  243  {10,48,80,105}
7  316  {10,50,56,200}
7  240  {10,50,75,105}
7  236  {10,55,66,105}
7  246  {10,56,60,120}
7  235  {10,56,65,104}
7  228  {10,56,72,90}
7  224  {10,60,70,84}
7  18284  {11,21,234,18018}
7  8124  {11,21,238,7854}
7  6432  {11,21,240,6160}
7  5356  {11,21,242,5082}
7  3056  {11,21,252,2772}
7  2144  {11,21,264,1848}
7  1632  {11,21,280,1320}
7  1404  {11,21,294,1078}
7  1264  {11,21,308,924}
7  1132  {11,21,330,770}
7  24058  {11,22,155,23870}
7  12201  {11,22,156,12012}
7  6274  {11,22,158,6083}
7  3736  {11,22,161,3542}
7  2508  {11,22,165,2310}
7  2049  {11,22,168,1848}
7  1441  {11,22,176,1232}
7  1216  {11,22,182,1001}
7  924  {11,22,198,693}
7  874  {11,22,203,638}
7  726  {11,22,231,462}
7  681  {11,22,252,396}
7  658  {11,22,275,350}
7  694  {11,23,154,506}
7  13069  {11,24,98,12936}
7  5678  {11,24,99,5544}
7  1460  {11,24,105,1320}
7  985  {11,24,110,840}
7  453  {11,24,154,264}
7  434  {11,24,168,231}
7  23220  {11,25,84,23100}
7  496  {11,25,110,350}
7  2116  {11,26,77,2002}
7  334  {11,26,143,154}
7  9649  {11,28,62,9548}
7  2874  {11,28,63,2772}
7  1029  {11,28,66,924}
7  424  {11,28,77,308}
7  354  {11,28,84,231}
7  289  {11,28,110,140}
7  4564  {11,29,58,4466}
7  10490  {11,30,54,10395}
7  2406  {11,30,55,2310}
7  1417  {11,30,56,1320}
7  342  {11,30,70,231}
7  256  {11,30,105,110}
7  451  {11,32,56,352}
7  1324  {11,33,48,1232}
7  364  {11,33,56,264}
7  264  {11,33,66,154}
7  1630  {11,35,44,1540}
7  226  {11,35,70,110}
7  2861  {11,36,42,2772}
7  784  {11,36,44,693}
7  217  {11,40,56,110}
7  196  {11,42,66,77}
7  199  {11,44,56,88}
7  32288  {12,18,254,32004}
7  21705  {12,18,255,21420}
7  16414  {12,18,256,16128}
7  11124  {12,18,258,10836}
7  9613  {12,18,259,9324}
7  8480  {12,18,260,8190}
7  7599  {12,18,261,7308}
7  5838  {12,18,264,5544}
7  5084  {12,18,266,4788}
7  4519  {12,18,268,4221}
7  4080  {12,18,270,3780}
7  3579  {12,18,273,3276}
7  3204  {12,18,276,2898}
7  2913  {12,18,279,2604}
7  2830  {12,18,280,2520}
7  2334  {12,18,288,2016}
7  2088  {12,18,294,1764}
7  1905  {12,18,300,1575}
7  1879  {12,18,301,1548}
7  1764  {12,18,306,1428}
7  1724  {12,18,308,1386}
7  1605  {12,18,315,1260}
7  1488  {12,18,324,1134}
7  1399  {12,18,333,1036}
7  1374  {12,18,336,1008}
7  1280  {12,18,350,900}
7  11350  {12,19,147,11172}
7  3375  {12,19,152,3192}
7  2693  {12,19,154,2508}
7  1263  {12,19,168,1064}
7  658  {12,19,228,399}
7  625  {12,19,252,342}
7  11268  {12,20,106,11130}
7  3920  {12,20,108,3780}
7  2452  {12,20,110,2310}
7  1824  {12,20,112,1680}
7  1476  {12,20,114,1330}
7  992  {12,20,120,840}
7  788  {12,20,126,630}
7  708  {12,20,130,546}
7  592  {12,20,140,420}
7  532  {12,20,150,350}
7  516  {12,20,154,330}
7  480  {12,20,168,280}
7  464  {12,20,180,252}
7  7258  {12,21,85,7140}
7  3731  {12,21,86,3612}
7  2556  {12,21,87,2436}
7  1969  {12,21,88,1848}
7  1383  {12,21,90,1260}
7  1216  {12,21,91,1092}
7  1091  {12,21,92,966}
7  994  {12,21,93,868}
7  801  {12,21,96,672}
7  719  {12,21,98,588}
7  658  {12,21,100,525}
7  611  {12,21,102,476}
7  558  {12,21,105,420}
7  519  {12,21,108,378}
7  481  {12,21,112,336}
7  433  {12,21,120,280}
7  411  {12,21,126,252}
7  396  {12,21,132,231}
7  394  {12,21,133,228}
7  383  {12,21,140,210}
7  376  {12,21,147,196}
7  371  {12,21,156,182}
7  5650  {12,22,72,5544}
7  1035  {12,22,77,924}
7  580  {12,22,84,462}
7  385  {12,22,99,252}
7  359  {12,22,105,220}
7  320  {12,22,132,154}
7  5894  {12,23,63,5796}
7  748  {12,23,69,644}
7  3285  {12,24,57,3192}
7  1718  {12,24,58,1624}
7  936  {12,24,60,840}
7  603  {12,24,63,504}
7  548  {12,24,64,448}
7  386  {12,24,70,280}
7  360  {12,24,72,252}
7  288  {12,24,84,168}
7  278  {12,24,88,154}
7  261  {12,24,105,120}
7  693  {12,25,56,600}
7  447  {12,25,60,350}
7  242  {12,25,100,105}
7  4454  {12,26,48,4368}
7  636  {12,26,52,546}
7  406  {12,26,56,312}
7  3864  {12,27,45,3780}
7  345  {12,27,54,252}
7  311  {12,27,56,216}
7  1889  {12,28,43,1806}
7  1008  {12,28,44,924}
7  715  {12,28,45,630}
7  569  {12,28,46,483}
7  424  {12,28,48,336}
7  383  {12,28,49,294}
7  329  {12,28,51,238}
7  283  {12,28,54,189}
7  264  {12,28,56,168}
7  240  {12,28,60,140}
7  229  {12,28,63,126}
7  215  {12,28,70,105}
7  209  {12,28,78,91}
7  24441  {12,29,40,24360}
7  895  {12,29,42,812}
7  1901  {12,30,39,1820}
7  922  {12,30,40,840}
7  504  {12,30,42,420}
7  339  {12,30,45,252}
7  218  {12,30,56,120}
7  207  {12,30,60,105}
7  196  {12,30,70,84}
7  2096  {12,32,36,2016}
7  310  {12,32,42,224}
7  196  {12,32,56,96}
7  1620  {12,33,35,1540}
7  774  {12,33,36,693}
7  243  {12,33,44,154}
7  189  {12,33,56,88}
7  398  {12,35,36,315}
7  255  {12,35,40,168}
7  229  {12,35,42,140}
7  177  {12,35,60,70}
7  216  {12,36,42,126}
7  198  {12,36,45,105}
7  176  {12,36,56,72}
7  168  {12,40,56,60}
7  175  {12,42,44,77}
7  30899  {13,16,294,30576}
7  9274  {13,16,301,8944}
7  7249  {13,16,304,6916}
7  4709  {13,16,312,4368}
7  3154  {13,16,325,2800}
7  2549  {13,16,336,2184}
7  1614  {13,17,156,1428}
7  5862  {13,18,98,5733}
7  1306  {13,18,105,1170}
7  694  {13,18,117,546}
7  439  {13,18,156,252}
7  7024  {13,19,76,6916}
7  16476  {13,20,63,16380}
7  1918  {13,20,65,1820}
7  294  {13,20,105,156}
7  2274  {13,21,56,2184}
7  294  {13,21,78,182}
7  274  {13,21,84,156}
7  14098  {13,22,49,14014}
7  2263  {13,24,42,2184}
7  249  {13,24,56,156}
7  624  {13,26,39,546}
7  354  {13,26,42,273}
7  199  {13,26,56,104}
7  239  {13,28,42,156}
7  184  {13,28,52,91}
7  44550  {14,15,211,44310}
7  22501  {14,15,212,22260}
7  15152  {14,15,213,14910}
7  11478  {14,15,214,11235}
7  9274  {14,15,215,9030}
7  7805  {14,15,216,7560}
7  6756  {14,15,217,6510}
7  5358  {14,15,219,5110}
7  4869  {14,15,220,4620}
7  4136  {14,15,222,3885}
7  3613  {14,15,224,3360}
7  3404  {14,15,225,3150}
7  2917  {14,15,228,2660}
7  2674  {14,15,230,2415}
7  2570  {14,15,231,2310}
7  2238  {14,15,235,1974}
7  2052  {14,15,238,1785}
7  1949  {14,15,240,1680}
7  1744  {14,15,245,1470}
7  1710  {14,15,246,1435}
7  1541  {14,15,252,1260}
7  1474  {14,15,255,1190}
7  1398  {14,15,259,1110}
7  1381  {14,15,260,1092}
7  1244  {14,15,270,945}
7  1212  {14,15,273,910}
7  1149  {14,15,280,840}
7  1112  {14,15,285,798}
7  1058  {14,15,294,735}
7  1029  {14,15,300,700}
7  997  {14,15,308,660}
7  990  {14,15,310,651}
7  974  {14,15,315,630}
7  925  {14,15,336,560}
7  904  {14,15,350,525}
7  12799  {14,16,113,12656}
7  6528  {14,16,114,6384}
7  3394  {14,16,116,3248}
7  2053  {14,16,119,1904}
7  1830  {14,16,120,1680}
7  1164  {14,16,126,1008}
7  1054  {14,16,128,896}
7  730  {14,16,140,560}
7  678  {14,16,144,504}
7  559  {14,16,161,368}
7  534  {14,16,168,336}
7  514  {14,16,176,308}
7  480  {14,16,210,240}
7  9631  {14,17,80,9520}
7  1543  {14,17,84,1428}
7  1306  {14,17,85,1190}
7  490  {14,17,102,357}
7  415  {14,17,112,272}
7  388  {14,17,119,238}
7  4128  {14,18,64,4032}
7  1484  {14,18,66,1386}
7  732  {14,18,70,630}
7  608  {14,18,72,504}
7  368  {14,18,84,252}
7  332  {14,18,90,210}
7  288  {14,18,112,144}
7  3678  {14,19,54,3591}
7  1153  {14,19,56,1064}
7  888  {14,19,57,798}
7  438  {14,19,63,342}
7  6661  {14,20,47,6580}
7  1762  {14,20,48,1680}
7  1063  {14,20,49,980}
7  784  {14,20,50,700}
7  541  {14,20,52,455}
7  397  {14,20,55,308}
7  370  {14,20,56,280}
7  304  {14,20,60,210}
7  277  {14,20,63,180}
7  244  {14,20,70,140}
7  226  {14,20,80,112}
7  223  {14,20,84,105}
7  1884  {14,21,43,1806}
7  1003  {14,21,44,924}
7  710  {14,21,45,630}
7  564  {14,21,46,483}
7  419  {14,21,48,336}
7  378  {14,21,49,294}
7  324  {14,21,51,238}
7  278  {14,21,54,189}
7  259  {14,21,56,168}
7  235  {14,21,60,140}
7  224  {14,21,63,126}
7  210  {14,21,70,105}
7  204  {14,21,78,91}
7  3078  {14,22,39,3003}
7  540  {14,22,42,462}
7  388  {14,22,44,308}
7  198  {14,22,63,99}
7  5869  {14,23,36,5796}
7  244  {14,23,46,161}
7  2928  {14,24,34,2856}
7  913  {14,24,35,840}
7  578  {14,24,36,504}
7  288  {14,24,40,210}
7  248  {14,24,42,168}
7  198  {14,24,48,112}
7  178  {14,24,56,84}
7  173  {14,24,63,72}
7  5671  {14,25,32,5600}
7  424  {14,25,35,350}
7  1016  {14,27,30,945}
7  158  {14,27,54,63}
7  883  {14,28,29,812}
7  492  {14,28,30,420}
7  298  {14,28,32,224}
7  217  {14,28,35,140}
7  204  {14,28,36,126}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
7  184  {14,30,35,105}
7  156  {14,30,42,70}
7  152  {14,30,45,63}
7  145  {14,35,40,56}
7  2906  {15,16,75,2800}
7  951  {15,16,80,840}
7  675  {15,16,84,560}
7  376  {15,16,105,240}
7  353  {15,16,112,210}
7  1520  {15,17,60,1428}
7  4492  {15,18,49,4410}
7  1658  {15,18,50,1575}
7  449  {15,18,56,360}
7  345  {15,18,60,252}
7  306  {15,18,63,210}
7  228  {15,18,90,105}
7  1894  {15,20,39,1820}
7  915  {15,20,40,840}
7  497  {15,20,42,420}
7  332  {15,20,45,252}
7  211  {15,20,56,120}
7  200  {15,20,60,105}
7  189  {15,20,70,84}
7  1332  {15,21,36,1260}
7  356  {15,21,40,280}
7  288  {15,21,42,210}
7  180  {15,21,60,84}
7  2380  {15,22,33,2310}
7  534  {15,22,35,462}
7  24428  {15,24,29,24360}
7  909  {15,24,30,840}
7  242  {15,24,35,168}
7  184  {15,24,40,105}
7  155  {15,24,56,60}
7  2168  {15,25,28,2100}
7  420  {15,25,30,350}
7  213  {15,28,30,140}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  150  {15,30,35,70}
7  141  {15,30,40,56}
7  1509  {16,17,48,1428}
7  361  {16,17,56,272}
7  1084  {16,18,42,1008}
7  334  {16,18,48,252}
7  234  {16,18,56,144}
7  209  {16,18,63,112}
7  18549  {16,20,33,18480}
7  631  {16,20,35,560}
7  189  {16,20,48,105}
7  172  {16,20,56,80}
7  741  {16,21,32,672}
7  312  {16,21,35,240}
7  191  {16,21,42,112}
7  169  {16,21,48,84}
7  4434  {16,24,26,4368}
7  404  {16,24,28,336}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  1499  {17,18,36,1428}
7  1495  {17,20,30,1428}
7  1494  {17,21,28,1428}
7  3845  {18,20,27,3780}
7  696  {18,20,28,630}
7  320  {18,20,30,252}
7  179  {18,20,36,105}
7  884  {18,21,26,819}
7  444  {18,21,27,378}
7  319  {18,21,28,252}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  5608  {18,22,24,5544}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  1128  {19,21,24,1064}
7  345  {20,21,24,280}
7  174  {20,21,28,105}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  129  {21,24,28,56}
7  120  {21,24,35,40}
解的个数为： 1037组

---

作者: ysr    时间: 2026-1-17 08:33
这回终于有了这一组解：7  612  {8,120,154,330}
改变参数居然增加了这么多组解！
怀疑拆分解的个数是不是有限的？？

代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(Text3)
a = Val(n)

Do While a >= Val(n) And a <= 6 * n
b = Val(a + 1)


Do While b <= Val(18 * n)
c = Val(b + 1)

Do While c <= Val(50 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = s2 & n & "  " & k & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 08:56
输入：7和119
输出：无解

可见拆分成4个单位分数之和的分母的和的最小值就是120.

---

作者: ysr    时间: 2026-1-17 09:28

王守恩 发表于 2026-1-16 20:39
代码A是7#的代码。代码B速度会快一些。\(M=\infty\ 改\ M=24\)——就是根据3#来的。又：网友指出{2,43,{5,6 ...

修改后的程序结果仍然不是一组解？少了不少！

输入：7
输出：
7  168  {12,40,56,60}
7  158  {14,27,54,63}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
7  156  {14,30,42,70}
7  152  {14,30,45,63}
7  145  {14,35,40,56}
7  155  {15,24,56,60}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  150  {15,30,35,70}
7  141  {15,30,40,56}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  129  {21,24,28,56}
7  120  {21,24,35,40}
解的个数为： 25组

需要比较大小，这次省力不少！

代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(n)

Do While a >= Val(n) And a <= 6 * n
b = Val(a + 1)


Do While b <= Val(18 * n)
c = Val(b + 1)

Do While c <= Val(50 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = s2 & n & "  " & k & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 09:34
又改了一下程序，这回解更少了，仍然需要比较大小？

输入：7
输出：
7  158  {14,27,54,63}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
7  155  {15,24,56,60}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  129  {21,24,28,56}
7  120  {21,24,35,40}
解的个数为： 19组

---

作者: ysr    时间: 2026-1-17 09:41
输入：2
输出：
2  48  {4,8,12,24}
2  43  {5,6,12,20}
解的个数为： 2组
修改前：输入：2和10000
输出：
2  179  {4,6,13,156}
2  108  {4,6,14,84}
2  85  {4,6,15,60}
2  74  {4,6,16,48}
2  64  {4,6,18,36}
2  161  {4,7,10,140}
2  65  {4,7,12,42}
2  53  {4,7,14,28}
2  93  {4,8,9,72}
2  62  {4,8,10,40}
2  48  {4,8,12,24}
2  139  {5,6,8,120}
2  65  {5,6,9,45}
2  51  {5,6,10,30}
2  43  {5,6,12,20}
解的个数为： 15组

---

作者: ysr    时间: 2026-1-17 09:42
修改后的结果：
输入：8
输出：
8  186  {16,30,60,80}
8  192  {16,32,48,96}
8  192  {18,24,60,90}
8  189  {18,24,63,84}
8  171  {18,27,54,72}
8  165  {18,28,56,63}
8  178  {18,30,40,90}
8  165  {18,30,45,72}
8  182  {18,32,36,96}
8  176  {19,24,57,76}
8  185  {20,22,55,88}
8  191  {20,24,42,105}
8  179  {20,24,45,90}
8  172  {20,24,48,80}
8  169  {20,24,50,75}
8  165  {20,24,55,66}
8  185  {20,25,40,100}
8  158  {20,28,40,70}
8  178  {20,30,32,96}
8  171  {20,30,33,88}
8  158  {20,30,36,72}
8  150  {20,30,40,60}
8  148  {20,30,42,56}
8  171  {21,24,42,84}
8  166  {21,24,44,77}
8  177  {21,28,32,96}
8  170  {21,28,33,88}
8  157  {21,28,36,72}
8  149  {21,28,40,60}
8  147  {21,28,42,56}
8  181  {22,24,36,99}
8  156  {22,24,44,66}
8  179  {24,25,30,100}
8  141  {24,26,39,52}
8  141  {24,27,36,54}
8  152  {24,28,30,70}
8  135  {24,30,36,45}
解的个数为： 37组

---

作者: ysr    时间: 2026-1-17 10:13

王守恩 发表于 2026-1-16 20:39
代码A是7#的代码。代码B速度会快一些。\(M=\infty\ 改\ M=24\)——就是根据3#来的。又：网友指出{2,43,{5,6 ...

这回有了这一组解：
输入：2
输出：
2  48  {4,8,12,24}
2  43  {4,9,12,18}
2  41  {4,10,12,15}
2  43  {5,6,12,20}
解的个数为： 4组

代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(n)

Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(5 * n)
c = Val(b + 1)

Do While c <= Val(10 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = s2 & n & "  " & k & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 10:14
8  190  {15,35,56,84}
8  185  {15,35,63,72}
8  183  {15,36,60,72}
8  190  {15,40,45,90}
8  183  {15,40,48,80}
8  180  {15,40,50,75}
8  176  {15,40,55,66}
8  186  {16,30,60,80}
8  192  {16,32,48,96}
8  185  {16,33,48,88}
8  177  {16,36,45,80}
8  172  {16,36,48,72}
8  164  {16,40,48,60}
8  176  {17,34,40,85}
8  192  {18,24,60,90}
8  189  {18,24,63,84}
8  171  {18,27,54,72}
8  165  {18,28,56,63}
8  178  {18,30,40,90}
8  165  {18,30,45,72}
8  182  {18,32,36,96}
8  175  {18,33,36,88}
8  156  {18,35,40,63}
8  154  {18,36,40,60}
8  152  {18,36,42,56}
8  176  {19,24,57,76}
8  185  {20,22,55,88}
8  191  {20,24,42,105}
8  179  {20,24,45,90}
8  172  {20,24,48,80}
8  169  {20,24,50,75}
8  165  {20,24,55,66}
8  185  {20,25,40,100}
8  158  {20,28,40,70}
8  178  {20,30,32,96}
8  171  {20,30,33,88}
8  158  {20,30,36,72}
8  150  {20,30,40,60}
8  148  {20,30,42,56}
8  141  {20,36,40,45}
8  171  {21,24,42,84}
8  166  {21,24,44,77}
8  177  {21,28,32,96}
8  170  {21,28,33,88}
8  157  {21,28,36,72}
8  149  {21,28,40,60}
8  147  {21,28,42,56}
8  138  {21,35,40,42}
8  181  {22,24,36,99}
8  156  {22,24,44,66}
8  179  {24,25,30,100}
8  141  {24,26,39,52}
8  141  {24,27,36,54}
8  152  {24,28,30,70}
8  135  {24,30,36,45}
解的个数为： 55组

---

作者: ysr    时间: 2026-1-17 10:16
7  158  {14,27,54,63}
7  168  {14,28,42,84}
7  163  {14,28,44,77}
7  156  {14,30,42,70}
7  152  {14,30,45,63}
7  145  {14,35,40,56}
7  155  {15,24,56,60}
7  162  {15,28,35,84}
7  145  {15,28,42,60}
7  150  {15,30,35,70}
7  141  {15,30,40,56}
7  144  {16,24,48,56}
7  134  {16,28,42,48}
7  164  {18,21,35,90}
7  159  {18,21,36,84}
7  144  {18,21,42,63}
7  134  {18,24,36,56}
7  124  {18,28,36,42}
7  155  {20,21,30,84}
7  136  {20,21,35,60}
7  130  {20,24,30,56}
7  120  {20,28,30,42}
7  129  {21,24,28,56}
7  120  {21,24,35,40}
解的个数为： 24组

---

作者: ysr    时间: 2026-1-17 10:21
代码已经修改为如下这样的：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(n)

Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(5 * n)
c = Val(b + 1)

Do While c <= Val(10 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = s2 & n & "  " & k & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 10:34
原理和方法仍然有待研究，网友的方法很有效（不过我仍然不太明白），其中的某些参数需要研究和修改。
经过验证1/7拆分成4个单位分数的和，其分母的最小和值是120，但，不懂计算原理，我仍然无法证明！

学习了，谢谢网友！给您点赞！

---

作者: ysr    时间: 2026-1-17 12:29
这回可以输出最小解了
输入：2
输出：
2  41  {4,10,12,15}
解的个数为： 3组

输入：7
输出：
7  120  {21,24,35,40}
解的个数为： 12组

输入：8
输出：
8  135  {24,30,36,45}
解的个数为： 20组

输入：9
输出：
9  153  {26,36,39,52}
解的个数为： 18组

输入：10
输出：
10  168  {30,40,42,56}
解的个数为： 22组

输入：11
输出：
11  180  {36,44,45,55}
解的个数为： 7组

是参考网友7#楼的代码，虽然不太懂

谢谢网友！！

稍加修改后的代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)

Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(6 * n)
c = Val(b + 1)

Do While c <= Val(10 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 13:06
如何把 1/7 拆分成四个不同单位分数的和，并且四个分母之和在某整数内为最大：
输入：7和50000
输出：
7  24441  {12,29,40,24360}
解中分母的和值不同的解的个数为： 7组

输入：7和20000
输出：
7  18549  {16,20,33,18480}
解中分母的和值不同的解的个数为： 8组

输入：7和90000
输出：
7  24441  {12,29,40,24360}
解中分母的和值不同的解的个数为： 7组

输入：7和210000
输出：
7  24441  {12,29,40,24360}
解中分母的和值不同的解的个数为： 7组

输入：7和310000
输出;
7  24441  {12,29,40,24360}
解中分母的和值不同的解的个数为： 7组

参数不变的情况下是有最大和值解的，求最大和值，参数也要变化

程序代码如下：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(6 * n)
c = Val(b + 1)

Do While c <= Val(10 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k >= m And k <= m1 Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解中分母的和值不同的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 13:14
输入：7和310000
输出：
7  145055  {9,46,100,144900}
需要比较的解的个数为： 3组

参数修改后的代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(10 * n)
c = Val(b + 1)

Do While c <= Val(20 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k >= m And k <= m1 Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 13:19
没有最大和值，即使某数内的最大和值也需要调整参数才能找到，否则照出来的也不一定对

---

作者: ysr    时间: 2026-1-17 13:27
输入：7和310000
输出：
7  306778  {9,34,429,306306}
需要比较的解的个数为： 3组

稍微改了一下参数，结果就不一样了，这是最大和值吗？不一定，找最大和值，没有意义了！（还是找最小值有意义）

代码如下：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(10 * n)
c = Val(b + 1)

Do While c <= Val(700 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k >= m And k <= m1 Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 13:50
13  216  {39,52,60,65}
需要比较的解的个数为： 9组
这也是四个分母的最小和值。

代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(10 * n)
c = Val(b + 1)

Do While c <= Val(50 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 16:13
输入：100
输出：
100  1611  {350,406,420,435}   不知道是否正确，需要验证一下！
需要比较的解的个数为： 66组

代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(5 * n)
c = Val(b + 1)

Do While c <= Val(10 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And Val(u - Int(u / v) * Abs(v)) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

---

作者: ysr    时间: 2026-1-17 16:36

ysr 发表于 2026-1-17 08:13
输入：100
输出：
100  1611  {350,406,420,435}   不知道是否正确，需要验证一下！

经过验证100  1611  {350,406,420,435}，这一组解是对的
也就是1/350+1/406+1/420+1/435=8526/852600=1/100

---

作者: ysr    时间: 2026-1-17 17:50
网友点评：”正确——{100, 1611, {350, 406, 420, 435}——7#代码有瑕疵。“
答：谢谢朋友的指导和鼓励！
7#的代码也不是有瑕疵，仅仅是参数需要调整而已，设a <b< c< d，我认为a≥n+1，a的最低值应该取到n+1，而不是仅仅2*n以上的数，
随着整数的增大，其他参数也可能需要微量调整。

---

作者: ysr    时间: 2026-1-17 17:58
99  1593  {352,396,416,429}
需要比较的解的个数为： 69组

参数调整以后枚举的范围可能是扩大了，时间会变长一些！范围扩大可能是更准确一点，但，我没明白原理无法证明是最低值，仅仅算是验证而已。
您的数据也很好用，很准确，比如设定M=24*n，就很好很管用，至于为什么这样做，我真的不知道！

---

作者: ysr    时间: 2026-1-17 20:40
网友点评：”原理:1=1/2+1/4+1/6+1/12——2+4+6+12=24——24是最高值。“
答：奥！原来是这么回事。
我也注意到了，你的数据中第一项n=1时，四个分母的和是24，很好！确实很有效，原来可以这样用，学习了!谢谢！

---

作者: 王守恩    时间: 2026-1-18 05:54
我喜欢一批一批的做题目。——来个“1”的拆分资料。

A213062——Minimal sum x(1) +...+ x(n) such that 1/x(1) +...+ 1/x(n) = 1, the x(i) being n distinct positive integers.

1, 0, 11, 24, 38, 50, 71, 87, 106, 127, 151, 185, 211, 249, 288, 325, 364, 406, 459, 508, 550, 613, 676, 728,——Table of n, a(n) for n=1..24.

a(3) = 11 = 2 + 3 + 6, because 1/2+1/3+1/6 is the only Egyptian fraction with 3 terms having 1 as sum.
a(4) = 24 = 2 + 4 + 6 + 12 is the smallest sum of denominators among the six 4-term Egyptian fractions equal to 1.
a(5) = 38 = 3 + 4 + 5 + 6 + 20, least sum of denominators among 72 possible 5-term Egyptian fractions equal to 1.
a(6) = 50 = 3 + 4 + 6 + 10 + 12 + 15, least sum of denominators among 2320 possible 6-term Egyptian fractions equal to 1.
a(7) <= 71 = 3 + 5 + 20 + 6 + 10 + 12 + 15 (obtained from n=6 using 1/4 = 1/5 + 1/20).
a(8) <= 114 = 3 + 5 + 20 + 7 + 42 + 10 + 12 + 15 (obtained using 1/6 = 1/7 + 1/42).
a(9) <= 145 = 3 + 6 + 30 + 20 + 7 + 42 + 10 + 12 + 15 (obtained using 1/5 = 1/6 + 1/30).
a(10) <= 202 = 3 + 6 + 30 + 20 + 8 + 56 + 42 + 10 + 12 + 15 (obtained using 1/7 = 1/8 + 1/56).

A213062——没有的——我来补上。

03, 011=2+3+6,
04, 024=2+4+6+12,——这是主帖——24。
05, 038=3+4+5+06+20,
06, 050=3+4+6+10+12+15,
07, 071=3+4+9+10+12+15+18,
08, 087=4+5+6+09+10+15+18+20,
09, 106=4+6+8+09+10+12+15+18+24,
10, 127=5+6+8+09+10+12+15+18+20+24,
11, 151=6+7+8+09+10+12+14+15+18+24+28,
12, 185=6+7+9+10+11+12+14+15+18+22+28+33,
13, 211=7+8+9+10+11+12+14+15+18+22+24+28+33,
14, 249=7+8+9+10+11+14+15+18+20+22+24+28+30+33,
15, 288=7+8+10+11+12+14+15+18+20+22+24+28+30+33+36,
16, 325=8+9+10+11+12+15+16+18+20+21+22+24+28+30+33+48,
17, 364=8+9+11+12+14+15+16+18+20+21+22+24+28+30+33+35+48,
18, 406=9+10+11+12+14+15+16+18+20+21+22+24+28+30+33+35+40+48,
19, 459=9+10+11+12+14+15+18+20+21+22+24+26+28+30+33+35+39+40+52,
20, 508=8+11+12+14+15+16+18+20+21+22+24+26+28+30+33+35+36+39+48+52,
21, 550=10+11+12+14+15+16+18+20+21+22+24+26+28+30+33+35+36+39+40+48+52,
22, 613=10+11+12+14+15+16+20+21+22+24+26+27+28+30+33+35+36+39+40+48+52+54,
23, 676=09+11+14+15+16+18+20+21+22+24+26+27+28+30+33+35+36+39+42+48+52+54+56,
24, 728=10+12+14+15+16+18+20+21+22+24+26+27+28+30+33+35+39+40+44+45+48+52+54+55,

---

作者: 王守恩    时间: 2026-1-18 06:01
这是“38”——原理没变。——答案有误。

1. LaunchKernels[]; ParallelTable[Module[{M = 38 n, S = {}, a, b, c, d, f, k, u, v}, For[a = 3 n, a ≤ 5 n, a++, For[b = a + 1, b ≤ 5 n, b++, For[c = b + 1, c < 6 n, c++,
   
2. For[d = c + 1, d < 8 n, d++, u = a*b*c*d*n; v = a*b*c*d - n (a*b*c + a*b*d + a*c*d + b*c*d);If[v > 0 && Mod[u, v] == 0, f = u/v;
   
3. If[d < f && f < 10 n, k = a + b + c + d + f; If[k ≤ M, M = k; S = {a, b, c, d, f}]]]]]]];{n, M, S}], {n, 50}]

复制代码

{{1,  38, {},{3, 4, 5, 6, 20——后加}},   {2,   58,   {6,    9,    10,   15,   18}},    {3,   86,   {10,   12,   15,   21,   28}},  {4,  111,   {12,  20,   21,   28,   30}},  {5,  133,   {20,   21,   24,  28,   40}},
{6, 160,   {21,  28,   30,   36,   45}},    {7,  181,  {28,  30,   36,   42,   45}},    {8,  209,  {30,   36,   42,   45,   56}},  {9,  234,   {36,  40,   42,   56,   60}},  {10, 259,  {40,   42,   55,  56,   66}},
{11, 284,  {44,  50,   55,   60,   75}},    {12, 306,  {50,  55,   60,   66,   75}},   {13, 361,  {45,   55,   66,   78,  117}}, {14, 360,  {56,  63,   72,   78,   91}},  {15, 383,  {60,   70,   78,  84,   91}},
{16, 403,  {70,  78,   80,   84,   91}},    {17, 440,  {68,  78,   84,   91,  119}},  {18, 459,  {72,   88,   90,   99,  110}}, {19, 483,  {78,   91,   95, 105, 114}},  {20, 507,  {88,   90,   99, 110,  120}},
{21, 539,  {80,  105, 110, 112, 132}},   {22, 561,  {88,  108, 110, 120, 135}},  {23, 587,  {92,   110, 115, 132, 138}}, {24, 605,  {108, 110, 120, 132, 135}}, {25, 635,  {108, 110, 132, 135, 150}},
{26, 656,  {117, 120, 130, 136, 153}},  {27, 691,  {108, 126, 136, 153, 168}},  {28, 713,  {112, 136, 144, 153, 168}}, {29, 759,  {112, 126, 144, 174, 203}}, {30, 763,  {126, 136, 153, 168, 180}},
{31, 833,  {120, 124, 155, 186, 248}},  {32, 806,  {140, 156, 160, 168, 182}},  {33, 835,  {147, 154, 156, 182, 196}}, {34, 857,  {144, 170, 176, 180, 187}}, {35, 879,  {156, 171, 180, 182, 190}},
{36, 904,  {168, 171, 175, 190, 200}},  {37, 967,  {148, 156, 182, 222, 259}},  {38, 961,  {168, 175, 190, 200, 228}}, {39, 982,  {176, 180, 198, 208, 220}}, {40, 1014, {176, 180, 198, 220, 240}},
{41, 1121, {156, 168, 182, 287, 328}}, {42, 1069, {180, 189, 216, 220, 264}}, {43, 1223, {140, 180, 215, 301, 387}}, {44, 1108, {198, 207, 220, 230, 253}}, {45, 1131, {207, 210, 230, 231, 253}},
{46, 1174, {184, 220, 230, 264, 276}}, {47, 1301, {171, 190, 235, 282, 423}}, {48, 1209, {210, 234, 240, 252, 273}}, {49, 1249, {196, 234, 252, 273, 294}}, {50, 1269, {210, 234, 252, 273, 300}}}

---

作者: 王守恩    时间: 2026-1-19 04:49
—这些应该没错了！——可惜OEIS没有这串数！

1. LaunchKernels[]; ParallelTable[Module[{M = 24 n, S = {}, a, b, c, d, k, u, v}, For[a = n + 1, a < 4 n, a++, For[b = a + 1, b ≤ 6 n, b++, For[c = b + 1, c ≤ 7 n, c++, u = a*b*c*n;
   
2. v = a*b*c - n (a*b + (a + b) c); If[v > 0 && Mod[u, v] == 0, d = u/v; If[c < d && d ≤ 12 n, k = a + b + c + d; If[k ≤ M, M = k; S = {a, b, c, d}]]]]]]; {n, M, S}], {n, 200}]

复制代码

  {{1,  24,   {2,    4,     6,   12}},     {2,   41,   {4,   10,   12,  15}},    {3,   52,   {9,  10,  15,   18}},      {4,  74,   {10,   15,   21,   28}},      {5,   84,   {15,  20,   21,  28}},
   {6,   99,  {20,  21,   28,  30}},     {7,  120,  {21,  24,   35,  40}},    {8,  135,  {24, 30,  36,   45}},      {9, 153,  {26,    36,   39,  52}},     {10, 168,  {30,  40,   42,  56}},
  {11, 180, {36,  44,   45,   55}},    {12, 198,  {40,  42,   56,  60}},   {13, 216,  {39, 52,  60,   65}},      {14, 229,  {45,   55,   63,  66}},     {15, 246,  {50,  55,   66,  75}},
  {16, 261, {55,  60,   66,   80}},    {17, 308,  {44,  68,   77, 119}},   {18, 297,  {60, 63,  84,   90}},      {19, 330,  {55,   66,   95, 114}},    {20, 323,  {70,  78,   84,  91}},
  {21, 344, {70,  78,    91, 105}},    {22, 360, {72,  88,   90, 110}},    {23, 405,  {60,  92, 115, 138}},     {24, 387,  {88,   90,   99, 110}},    {25, 417,  {75, 100, 110, 132}},
  {26, 420, {91,  104, 105, 120}},   {27, 440, {90, 108, 110, 132}},    {28, 458,  {90, 110, 126, 132}},     {29, 546,  {78,   91, 174, 203}},    {30, 485, {108, 110, 132, 135}},
  {31, 626, {72,  120, 155, 279}},   {32, 522, {110,120, 132, 160}},    {33, 536, {112, 126, 144, 154}},    {34, 583, {117, 119, 126, 221}},    {35, 576, {112, 140, 144, 180}},
  {36, 583, {126, 136, 153, 168}},   {37, 713, {84, 148, 222, 259}},    {38, 612, {136, 152, 153, 171}},    {39, 641, {136, 144, 153, 208}},    {40, 646, {140, 156, 168, 182}},
  {41, 776, {120, 123, 205, 328}},   {42, 681, {147, 156, 182, 196}},   {43, 906, {129, 132, 172, 473}},    {44, 706, {165, 171, 180, 190}},    {45, 723, {168, 175, 180, 200}},
  {46, 775, {156, 161, 182, 276}},   {47,1008,{112, 144, 329, 423}},    {48, 774, {176, 180, 198, 220}},    {49, 824, {147, 180, 245, 252}},    {50, 817, {175, 180, 210, 252}},
  {51, 839, {176, 187, 204, 272}},    {52, 840, {182, 208, 210, 240}},   {53, 1189, {120, 168, 371, 530}},   {54, 880, {180, 216, 220, 264}},    {55, 888, {198, 207, 230, 253}},
  {56, 912, {189, 216, 234, 273}},    {57, 925, {190, 228, 247, 260}},   {58, 1051, {145, 230, 299, 377}},   {59, 1386, {126, 198, 413, 649}},   {60, 969, {210, 234, 252, 273}},
  {61, 1279, {120, 305, 366, 488}},   {62, 1003, {217, 234, 273, 279}},  {63, 1012, {231, 252, 253, 276}},  {64, 1044, {220, 240, 264, 320}},  {65, 1052, {234, 240, 272, 306}},
  {66, 1067, {234, 252, 273, 308}},   {67, 1374, {168, 201, 469, 536}},  {68, 1107, {240, 255, 272, 340}},  {69, 1151, {208, 276, 299, 368}},  {70, 1125, {260, 273, 280, 312}},
  {71, 1631, {140, 284, 497, 710}},   {72, 1155, {270, 280, 297, 308}},  {73, 1732, {126, 438, 511, 657}},  {74, 1205, {259, 280, 296, 370}},  {75, 1217, {252, 300, 315, 350}},
  {76, 1224, {272, 304, 306, 342}},   {77, 1247, {264, 308, 315, 360}},  {78, 1260, {273, 312, 315, 360}},  {79, 1602, {180, 316, 395, 711}},  {80, 1291, {285, 304, 342, 360}},
  {81, 1300, {300, 324, 325, 351}},   {82, 1363, {275, 287, 350, 451}},  {83, 1992, {166, 332, 498, 996}},  {84, 1351, {312, 315, 360, 364}},  {85, 1368, {306, 340, 342, 380}},
  {86, 1518, {253, 276, 473, 516}},   {87, 1430, {286, 319, 390, 435}},  {88, 1412, {330, 342, 360, 380}},  {89, 1990, {210, 267, 623, 890}},  {90, 1446, {336, 350, 360, 400}},
  {91, 1487, {308, 330, 420, 429}},   {92, 1481, {342, 345, 380, 414}},   {93, 1549, {310, 340, 372, 527}},  {94, 1599, {282, 330, 470, 517}},  {95, 1532, {330, 385, 399, 418}},
  {96, 1548, {352, 360, 396, 440}},   {97, 2328, {194, 388, 582, 1164}},  {98, 1583, {342, 380, 420, 441}},  {99,1593, {352, 396, 416, 429}},  {100, 1611, {350, 406, 420, 435}},
{101, 2240, {220, 404, 505, 1111}}, {102, 1643, {360, 408, 425, 450}}, {103, 2064, {210, 515, 618, 721}}, {104, 1677, {360, 425, 442, 450}}, {105, 1686, {390, 406, 435, 455}},
{106, 1907, {264, 424,  583,  636}}, {107, 2568, {214, 428, 642,1284}}, {108, 1741, {377, 432, 464, 468}}, {109, 2616, {218, 436, 654,1308}}, {110, 1765, {416, 429, 440, 480}},
{111, 1807, {364, 444,  481,  518}}, {112, 1802, {406, 435, 465,  496}}, {113, 2712, {226, 452, 678,1356}}, {114, 1836, {408, 456, 459, 513}}, {115, 1848, {420, 460, 462, 506}},
{116, 1860, {435, 464,  465,  496}}, {117, 1884, {429, 440, 495,  520}}, {118, 1953, {360, 472, 531, 590}}, {119, 1920, {420, 476, 480, 544}}, {120, 1929, {440, 465, 496, 528}}}
{121, 1971, {420, 462,  484,  605}}, {122, 1968, {427, 488, 504,  549}}, {123, 2027, {410, 465, 496, 656}}, {124, 2006, {434, 468, 546, 558}}, {125, 2085, {375, 500, 550, 660}},
{126, 2024, {462, 504,  506,  552}}, {127, 3048, {254, 508, 762,1524}}, {128, 2081, {465, 480, 496, 640}}, {129, 2172, {430, 473, 495, 774}}, {130, 2091, {468, 513, 540, 570}},
{131, 3144, {262, 524,  786,1572}}, {132, 2117, {490, 528, 539,  560}}, {133, 2139, {494, 520, 525, 600}}, {134, 2523, {312, 536, 804, 871}}, {135, 2169, {504, 525, 540, 600}},
{136, 2196, {476, 525,  595,  600}}, {137, 3288, {274, 548, 822,1644}}, {138, 2245, {460, 550, 575, 660}}, {139, 3336, {278, 556, 834,1668}}, {140, 2244, {528, 560, 561, 595}},
{141, 2264, {517, 564,  572,  611}}, {142, 2559, {426, 429, 781, 923}}, {143, 2291, {540, 572, 585, 594}}, {144, 2310, {540, 560, 594,  616}}, {145, 2372, {493, 520, 663, 696}},
{146, 2537, {420, 511,  730,  876}}, {147, 2377, {539, 550, 588, 700}}, {148, 2410, {518, 560, 592, 740}}, {149, 3260, {280, 745,1043,1192}}, {150, 2411, {546, 585, 630, 650}},
{151, 3624, {302, 604,  906,1812}}, {152, 2448, {544, 608, 612, 684}}, {153, 2475, {520, 612, 663, 680}}, {154, 2479, {561, 595,  630,  693}}, {155, 2516, {527, 620, 629, 740}},
{156, 2500, {595, 612,  630,  663}}, {157, 3768, {314, 628, 942,1884}}, {158, 2911, {462, 474, 869,1106}}, {159, 2644, {477, 630, 742, 795}}, {160, 2582, {550, 660, 672, 700}},
{161, 2600, {550, 660,  690,  700}}, {162, 2600, {600, 648, 650,  702}}, {163, 3912, {326, 652, 978,1956}}, {164, 2660, {560, 656, 665, 779}}, {165, 2654, {605, 630, 693, 726}},
{166, 2984, {415, 660,  913,  996}}, {167, 4008, {334, 668,1002,2004}}, {168, 2702, {624, 630, 720, 728}}, {169, 2808, {507, 676, 780, 845}}, {170, 2731, {630, 660, 693, 748}},
{171, 2762, {608, 672,  684,  798}}, {172, 2864, {516, 714,  731, 903}}, {173, 4152, {346, 692,1038,2076}}, {174, 2831, {630, 638, 693, 870}}, {175, 2808, {650, 700, 702, 756}},
{176, 2824, {660, 684,  720,  760}}, {177, 3004, {585, 590, 767,1062}}, {178, 3441, {504, 623, 712,1602}}, {179, 4268, {330, 895,1074,1969}}, {180, 2889, {680, 684, 760, 765}},
{181, 4344, {362, 724,1086,2172}}, {182, 2925, {660, 715,  770, 780}}, {183, 2971, {610, 728, 793,  840}}, {184, 2962, {684, 690, 760,  828}}, {185, 2964, {703, 740, 741, 780}},
{186, 3009, {651, 702,  819, 837}}, {187, 3001, {684, 760,  765,  792}}, {188, 3023, {705, 720, 752,  846}}, {189, 3036, {693, 756, 759,  828}}, {190, 3060, {680, 760, 765, 855}},
{191, 4584, {382, 764,1146,2292}}, {192, 3083, {704, 759,  792, 828}}, {193, 4632, {386, 772,1158,2316}}, {194, 3415, {485, 873, 990,1067}}, {195, 3139, {702, 756, 820, 861}},
{196, 3147, {714, 784,  816, 833}}, {197, 4728, {394, 788,1182,2364}}, {198, 3186, {704, 792,  832, 858}}, {199, 4776, {398, 796,1194,2388}}, {200, 3204, {759, 792, 825, 828}}}

---

作者: 王守恩    时间: 2026-1-19 14:18


回头做这题目——会感觉简单了。

1. LaunchKernels[]; ParallelTable[Module[{M = 11 n, S = {2 n, 3 n, 6 n}, a, b, c, k, u, v}, For[a = n + 1, a < 3 n, a++,  For[b = a + 1, b < 4 n, b++, u = a*b*n; v = a*b - n (a + b);
   
2. v = a*b - n (a + b); If[v > 0 && Mod[u, v] == 0, c = u/v; If[b < c && c ≤ 6 n, k = a + b + c;   If[k ≤ M, M = k; S = {a, b, c}]]]]]; {n, M, S}], {n, 500}]

复制代码

{{ 1,     11,   {  2,     3,     6  }}, {  2,   22,  {  4,    6,     12 }}, {  3,   31,  {  6,   10,    15 }}, {  4,   37,  { 10,   12,   15 }}, {  5,   47,  { 12,  15,  20}},
{  6,     61,   { 12,   21,   28 }}, {  7,   71,  { 15,  21,    35 }}, {  8,   73,  { 21,  24,    28 }}, {  9,   85,  { 21,   28,   36 }}, { 10,  94,  { 24,  30,   40}},
{ 11,   119,  { 20,   44,   55 }}, { 12, 111, { 30,  36,    45 }}, { 13, 143, { 26,   39,   78 }}, { 14, 131, { 35,   40,   56 }}, { 15, 138, { 40,  42,   56}},
{ 16,   146,  { 42,   48,   56 }}, { 17, 187, { 34,  51,   102}}, { 18, 166, { 45,   55,   66 }}, { 19, 209, { 38,   57,  114}}, { 20, 181, { 55,  60,   66}},
{ 21,   191,  { 56,   63,   72 }}, { 22, 202, { 55,  70,    77 }}, { 23, 253, { 46,   69,  138}}, { 24, 219, { 63,   72,   84 }}, { 25, 235, { 60,  75,  100}},
{ 26,   251,  { 56,   91,  104}}, { 27, 255, { 63,  84,   108}}, { 28, 253, { 78,    84,  91 }}, { 29, 319, { 58,   87,  174}}, { 30, 274, { 78,  91,  105}},
{ 31,   341,  { 62,   93,  186}}, { 32, 292, { 84,  96,   112}}, { 33, 299, { 90,    99, 110}}, { 34, 328, { 85,   90,  153}}, { 35, 321, { 90, 105, 126}},
{ 36,   332,  { 90,  110, 132}}, { 37, 407, { 74, 111,  222}}, { 38, 395, { 72,  152, 171}}, { 39, 354, {104, 120, 130}}, { 40, 362, {110, 120, 132}},
{ 41,   451,  { 82,  123, 246}}, { 42, 382, {112, 126, 144}}, { 43, 473, { 86,  129, 258}}, { 44, 404, {110, 140, 154}}, { 45, 409, {120, 136, 153}},
{ 46,   478,  {110, 115, 253}}, { 47, 517, {  94, 141, 282}}, { 48, 433, {136, 144, 153}}, { 49, 497, {105, 147, 245}}, { 50, 463, {120, 168, 175}},
{ 51,   491,  {117, 153, 221}}, { 52, 481, {130, 156, 195}}, { 53, 583, {106, 159, 318}}, { 54, 496, {135, 171, 190}}, { 55, 500, {144, 176, 180}},
{ 56,   506,  {156, 168, 182}}, { 57, 533, {133, 190, 210}}, { 58, 638, {116, 174, 348}}, { 59, 649, {118, 177, 354}}, { 60, 541, {171, 180, 190}},
{ 61,   671,  {122, 183, 366}}, { 62, 622, {126, 217, 279}}, { 63, 571, {171, 190, 210}}, { 64, 584, {168, 192, 224}}, { 65, 587, {182, 195, 210}},
{ 66,   598,  {180, 198, 220}}, { 67, 737, {134, 201, 402}}, { 68, 629, {170, 204, 255}}, { 69, 645, {161, 231, 253}}, { 70, 642, {180, 210, 252}},
{ 71,   781,  {142, 213, 426}}, { 72, 657, {189, 216, 252}}, { 73, 803, {146, 219, 438}}, { 74, 814, {148, 222, 444}}, { 75, 684, {200, 220, 264}},
{ 76,   697,  {190, 247, 260}}, { 77, 705, {198, 234, 273}}, { 78, 708, {208, 240, 260}}, { 79, 869, {158, 237, 474}}, { 80, 724, {220, 240, 264}},
{ 81,   765,  {189, 252, 324}}, { 82, 892, {154, 287, 451}}, { 83, 913, {166, 249, 498}}, { 84, 759, {234, 252, 273}}, { 85, 767, {240, 255, 272}},
{ 86,   946,  {172, 258, 516}}, { 87, 899, {174, 290, 435}}, { 88, 793, {253, 264, 276}}, { 89, 979, {178, 267, 534}}, { 90, 818, {240, 272, 306}},
{ 91,   827,  {252, 260, 315}}, { 92, 851, {230, 276, 345}}, { 93, 853, {248, 264, 341}}, { 94, 1034, {188, 282, 564}}, {95, 863, {255, 285, 323}},
{ 96,   866,  {272, 288, 306}}, { 97, 1067, {194, 291, 582}}, {98, 917, {245, 280, 392}}, { 99,  897, {270, 297, 330}}, {100, 905, {275, 300, 330}},
{101, 1111, {202, 303, 606}}, {102, 960,  {238, 323, 399}}, {103, 1133, {206, 309, 618}}, {104, 937, {300, 312, 325}}, {105, 955, {280, 315, 360}},
{106, 1166, {212, 318, 636}}, {107, 1177, {214, 321, 642}}, {108,  976,  {300, 325, 351}}, {109, 1199, {218, 327, 654}}, {110, 1000, {288, 352, 360}},
{111, 1147, {222, 370, 555}}, {112, 1011, {315, 336, 360}}, {113, 1243, {226, 339, 678}}, {114, 1039, {304, 336, 399}}, {115, 1051, {315, 322, 414}},
{116, 1073, {290, 348, 435}}, {117, 1062, {312, 360, 390}}, {118, 1298, {236, 354, 708}}, {119, 1079, {323, 357, 399}}, {120, 1082, {342, 360, 380}},
{121, 1309, {220, 484, 605}}, {122, 1342, {244, 366, 732}}, {123, 1173, {312, 328, 533}}, {124, 1147, {310, 372, 465}}, {125, 1175, {300, 375, 500}},
{126, 1142, {342, 380, 420}}, {127, 1397, {254, 381, 762}}, {128, 1168, {336, 384, 448}}, {129, 1325, {301, 336, 688}}, {130, 1174, {364, 390, 420}},
{131, 1441, {262, 393, 786}}, {132, 1196, {360, 396, 440}}, {133, 1199, {380, 399, 420}}, {134, 1474, {268, 402, 804}}, {135, 1219, {378, 406, 435}},
{136, 1235, {360, 425, 450}}, {137, 1507, {274, 411, 822}}, {138, 1290, {322, 462, 506}}, {139, 1529, {278, 417, 834}}, {140, 1261, {406, 420, 435}},
{141, 1317, {330, 470, 517}}, {142, 1562, {284, 426, 852}}, {143, 1293, {396, 429, 468}}, {144, 1299, {408, 432, 459}}, {145, 1363, {348, 435, 580}},
{146, 1606, {292, 438, 876}}, {147, 1331, {406, 435, 490}}, {148, 1363, {364, 481, 518}}, {149, 1639, {298, 447, 894}}, {150, 1361, {400, 465, 496}},
{151, 1661, {302, 453, 906}}, {152, 1380, {408, 459, 513}}, {153, 1385, {425, 450, 510}}, {154, 1410, {396, 468, 546}}, {155, 1457, {372, 465, 620}},
{156, 1416, {416, 480, 520}}, {157, 1727, {314, 471, 942}}, {158, 1738, {316, 474, 948}}, {159, 1643, {318, 530, 795}}, {160, 1441, {465, 480, 496}},
{161, 1451, {462, 483, 506}}, {162, 1488, {405, 513, 570}}, {163, 1793, {326, 489, 978}}, {164, 1517, {410, 492, 615}}, {165, 1489, {465, 496, 528}},
{166, 1826, {332, 498, 996}}, {167, 1837, {334, 501, 1002}}, {168, 1518, {468, 504, 546}}, {169, 1859, {338, 507, 1014}}, {170, 1534, {480, 510, 544}},
{171, 1546, {468, 532, 546}}, {172, 1591, {430, 516,  645 }}, {173, 1903, {346, 519, 1038}}, {174, 1609, {420, 580, 609}}, {175, 1579, {490, 539, 550}},
{176, 1586, {506, 528, 552}}, {177, 1829, {354, 590,  885 }}, {178, 1958, {356, 534, 1068}}, {179, 1969, {358, 537, 1074}}, {180, 1623, {513, 540, 570}},
{181, 1991, {362, 543, 1086}}, {182, 1645, {520, 525, 600}}, {183, 1793, {390, 610,  793 }}, {184, 1679, {483, 552,  644 }}, {185, 1670, {518, 560, 592}},
{186, 1706, {496, 528,  682 }}, {187, 1726, {476, 550, 700}}, {188, 1700, {517, 572,  611 }}, {189, 1713, {513, 570,  630 }}, {190, 1726, {510, 570, 646}},
{191, 2101, {382, 573, 1146}}, {192, 1732, {544, 576, 612}}, {193, 2123, {386, 579, 1158}}, {194, 2134, {388, 582, 1164}}, {195, 1761, {546, 585, 630}},
{196, 1771, {546, 588,  637 }}, {197, 2167, {394, 591, 1182}}, {198, 1786, {561, 595, 630}}, {199, 2189, {398, 597, 1194}}, {200, 1810, {550, 600, 660}},
{201, 2077, {402, 670, 1005}}, {202, 2222, {404, 606, 1212}}, {203, 1853, {522, 638, 693}}, {204, 1837, {595, 612,  630 }}, {205, 1927, {492, 615, 820}},
{206, 2266, {412, 618, 1236}}, {207, 1871, {575, 621,  675 }}, {208, 1874, {600, 624, 650}}, {209, 1893, {570, 630,  693 }}, {210, 1910, {560, 630, 720}},
{211, 2321, {422, 633, 1266}}, {212, 1961, {530, 636,  795 }}, {213, 2133, {429, 781, 923}}, {214, 2354, {428, 642, 1284}}, {215, 2021, {516, 645, 860}},
{216, 1952, {600, 650,  702 }}, {217, 2003, {546, 651,  806 }}, {218, 2398, {436, 654, 1308}}, {219, 2263, {438, 730, 1095}}, {220, 1983, {630, 660, 693}},
{221, 2003, {595, 680,  728 }}, {222, 2037, {592, 629,  816 }}, {223, 2453, {446, 669, 1338}}, {224, 2022, {630, 672,  720 }}, {225, 2027, {650, 675, 702}},
{226, 2486, {452, 678, 1356}}, {227, 2497, {454, 681, 1362}}, {228, 2053, {666, 684,  703 }}, {229, 2519, {458, 687, 1374}}, {230, 2082, {630, 693, 759}},
{231, 2087, {637, 715,  735 }}, {232, 2117, {609, 696,  812 }}, {233, 2563, {466, 699, 1398}}, {234, 2110, {666, 703,  741 }}, {235, 2179, {611, 658, 910}},
{236, 2183, {590, 708,  885 }}, {237, 2437, {462, 869, 1106}}, {238, 2158, {646, 714,  798 }}, {239, 2629, {478, 717, 1434}}, {240, 2164, {684, 720, 760}},
{241, 2651, {482, 723, 1446}}, {242, 2222, {605, 770,  847 }}, {243, 2295, {567, 756,  972 }}, {244, 2246, {660, 671,  915 }}, {245, 2247, {630, 735, 882}},
{246, 2346, {624, 656, 1066}}, {247, 2277, {624, 741,  912 }}, {248, 2263, {651, 744,  868 }}, {249, 2573, {498, 830, 1245}}, {250, 2315, {600, 840, 875}},
{251, 2761, {502, 753, 1506}}, {252, 2277, {702, 756,  819 }}, {253, 2303, {663, 782,  858 }}, {254, 2794, {508, 762, 1524}}, {255, 2301, {720, 765, 816}},
{256, 2336, {672, 768,  896 }}, {257, 2827, {514, 771, 1542}}, {258, 2348, {714, 731,  903 }}, {259, 2365, {700, 740,  925 }}, {260, 2348, {728, 780, 840}},
{261, 2351, {756, 783,  812 }}, {262, 2882, {524, 786, 1572}}, {263, 2893, {526, 789, 1578}}, {264, 2379, {759, 792,  828 }}, {265, 2491, {636, 795, 1060}},
{266, 2398, {760, 798,  840 }}, {267, 2759, {534, 890, 1335}}, {268, 2479, {670, 804, 1005}}, {269, 2959, {538, 807, 1614}}, {270, 2437, {756, 820,  861 }},
{271, 2981, {542, 813, 1626}}, {272, 2470, {720, 850,  900 }}, {273, 2461, {780, 820,  861 }}, {274, 3014, {548, 822, 1644}}, {275, 2487, {759, 828,  900 }},
{276, 2486, {805, 820,  861 }}, {277, 3047, {554, 831, 1662}}, {278, 3058, {556, 834, 1668}}, {279, 2519, {783, 837,  899 }}, {280, 2521, {820, 840,  861 }},
{281, 3091, {562, 843, 1686}}, {282, 2557, {752, 893,  912 }}, {283, 3113, {566, 849, 1698}}, {284, 2627, {710, 852, 1065}}, {285, 2589, {765, 855,  969 }},
{286, 2586, {792, 858,  936 }}, {287, 2735, {644, 943, 1148}}, {288, 2598, {816, 864,  918 }}, {289, 3179, {578, 867, 1734}}, {290, 2669, {725, 900, 1044}},
{291, 3007, {582, 970, 1455}}, {292, 2701, {730, 876, 1095}}, {293, 3223, {586, 879, 1758}}, {294, 2661, {820, 861,  980 }}, {295, 2726, {720, 944, 1062}},
{296, 2701, {777, 888, 1036}}, {297, 2691, {810, 891,  990 }}, {298, 3278, {596, 894, 1788}}, {299, 2703, {828, 900,  975 }}, {300, 2705, {855, 900,  950 }},
{301, 2732, {840, 860, 1032}}, {302, 3322, {604, 906, 1812}}, {303, 3131, {606, 1010, 1515}}, {304, 2760, {816, 918, 1026}}, {305, 2867, {732, 915, 1220}},
{306, 2759, {884, 900,  975 }}, {307, 3377, {614, 921, 1842}}, {308, 2773, {903,  924,   946 }}, {309, 3193, {618, 1030, 1545}}, {310, 2914, {744, 930, 1240}},
{311, 3421, {622, 933, 1866}}, {312, 2811, {900, 936,  975 }}, {313, 3443, {626,  939,  1878}}, {314, 3454, {628,  942,  1884}}, {315, 2839, {903, 946,  990 }},
{316, 2923, {790, 948, 1185}}, {317, 3487, {634, 951, 1902}}, {318, 2918, {798, 1007, 1113}}, {319, 2879, {899,  957,  1023}}, {320, 2882, {930, 960,  992 }},
{321, 3317, {642, 1070, 1605}}, {322, 2902, {924, 966, 1012}}, {323, 2913, {918, 969, 1026}}, {324, 2928, {900,  975,  1053}}, {325, 2935, {910, 975, 1050}},
{326, 3586, {652,  978,  1956}}, {327, 3379, {654, 1090, 1635}}, {328, 2993, {861, 984, 1148}}, {329, 3046, {846, 940, 1260}}, {330, 2978, {930, 992, 1056}},
{331, 3641, {662,  993,  1986}}, {332, 3071, {830,  996,  1245}}, {333, 3047, {864, 999, 1184}}, {334, 3674, {668, 1002, 2004}}, {335, 3149, {804, 1005, 1340}},
{336, 3031, {952, 1008, 1071}}, {337, 3707, {674, 1011, 2022}}, {338, 3263, {728, 1183, 1352}}, {339, 3503, {678, 1130, 1695}}, {340, 3068, {960, 1020, 1088}},
{341, 3071, {992, 1023, 1056}}, {342, 3087, {952, 1064, 1071}}, {343, 3479, {735, 1029, 1715}}, {344, 3139, {903, 1032, 1204}}, {345, 3129, {920, 1081, 1128}},
{346, 3806, {692, 1038, 2076}}, {347, 3817, {694, 1041, 2082}}, {348, 3198, {899, 1023, 1276}}, {349, 3839, {698, 1047, 2094}}, {350, 3158, {980, 1078, 1100}},
{351, 3186, {936, 1080, 1170}}, {352, 3172, {1012, 1056, 1104}}, {353, 3883, {706, 1059, 2118}}, {354, 3203, {1003, 1020, 1180}}, {355, 3235, {923, 1105, 1207}},
{356, 3293, {890, 1068, 1335}}, {357, 3217, {1020, 1092, 1105}}, {358, 3938, {716, 1074, 2148}}, {359, 3949, { 718,  1077, 2154}}, {360, 3244, {1035, 1081, 1128}},
{361, 3971, {722, 1083, 2166}}, {362, 3982, { 724,  1086, 2172}}, {363, 3289, {990, 1089, 1210}}, {364, 3279, {1053, 1092, 1134}}, {365, 3403, { 876,  1140, 1387}},
{366, 3586, {780, 1220, 1586}}, {367, 4037, { 734,  1101, 2202}}, {368, 3313, {1081, 1104, 1128}}, {369, 3419, {918, 1107, 1394}}, {370, 3340, {1036, 1120, 1184}},
{371, 3671, {756, 1431, 1484}}, {372, 3375, { 988,  1178, 1209}}, {373, 4103, { 746,  1119, 2238}}, {374, 3395, {1020, 1100, 1275}}, {375, 3401, {1000, 1176, 1225}},
{376, 3400, {1034, 1144, 1222}}, {377, 3405, {1053, 1134, 1218}}, {378, 3426, {1026, 1140, 1260}}, {379, 4169, { 758,  1137, 2274}}, {380, 3439, {1045, 1140, 1254}},
{381, 3937, { 762,  1270, 1905}}, {382, 4202, { 764,  1146, 2292}}, {383, 4213, { 766,  1149, 2298}}, {384, 3464, {1088, 1152, 1224}}, {385, 3467, {1122, 1155, 1190}},
{386, 4246, { 772,  1158, 2316}}, {387, 3611, { 945,  1161, 1505}}, {388, 3589, { 970,  1164, 1455}}, {389, 4279, { 778,  1167, 2334}}, {390, 3522, {1092, 1170, 1260}},
{391, 3573, {1020, 1173, 1380}}, {392, 3542, {1092, 1176, 1274}}, {393, 4061, { 786,  1310, 1965}}, {394, 4334, { 788,  1182, 2364}}, {395, 3619, {1027, 1170, 1422}},
{396, 3572, {1122, 1190, 1260}}, {397, 4367, { 794,  1191, 2382}}, {398, 4378, { 796,  1194, 2388}}, {399, 3597, {1140, 1197, 1260}}, {400, 3601, {1176, 1200, 1225}},
{401, 4411, { 802,  1203, 2406}}, {402, 3717, {1072, 1104, 1541}}, {403, 3659, {1085, 1209, 1365}}, {404, 3737, {1010, 1212, 1515}}, {405, 3657, {1134, 1218, 1305}},
{406, 3706, {1044, 1276, 1386}}, {407, 3671, {1155, 1221, 1295}}, {408, 3674, {1190, 1224, 1260}}, {409, 4499, { 818,  1227, 2454}}, {410, 3827, {1080, 1107, 1640}},
{411, 4247, { 822,  1370, 2055}}, {412, 3811, {1030, 1236, 1545}}, {413, 3833, {1001, 1298, 1534}}, {414, 3742, {1150, 1242, 1350}}, {415, 3763, {1162, 1190, 1411}},
{416, 3748, {1200, 1248, 1300}}, {417, 4309, { 834,  1390, 2085}}, {418, 3785, {1140, 1265, 1380}}, {419, 4609, { 838,  1257, 2514}}, {420, 3783, {1218, 1260, 1305}},
{421, 4631, { 842,  1263, 2526}}, {422, 4642, { 844,  1266, 2532}}, {423, 3951, { 990,  1410, 1551}}, {424, 3869, {1113, 1272, 1484}}, {425, 3827, {1235, 1292, 1300}},
{426, 4266, { 858,  1562, 1846}}, {427, 4018, { 976,  1456, 1586}}, {428, 3959, {1070, 1284, 1605}}, {429, 3879, {1188, 1287, 1404}}, {430, 3958, {1075, 1333, 1550}},
{431, 4741, { 862,  1293, 2586}}, {432, 3897, {1224, 1296, 1377}}, {433, 4763, { 866,  1299, 2598}}, {434, 3918, {1218, 1305, 1395}}, {435, 3921, {1265, 1276, 1380}},
{436, 4033, {1090, 1308, 1635}}, {437, 3957, {1197, 1311, 1449}}, {438, 3987, {1140, 1387, 1460}}, {439, 4829, { 878,  1317, 2634}}, {440, 3965, {1265, 1320, 1380}},
{441, 3993, {1218, 1305, 1470}}, {442, 3982, {1287, 1309, 1386}}, {443, 4873, { 886,  1329, 2658}}, {444, 4021, {1221, 1320, 1480}}, {445, 4183, {1068, 1335, 1780}},
{446, 4906, { 892,  1338, 2676}}, {447, 4619, { 894,  1490, 2235}}, {448, 4044, {1260, 1344, 1440}}, {449, 4939, { 898,  1347, 2694}}, {450, 4054, {1300, 1350, 1404}},
{451, 4091, {1221, 1353, 1517}}, {452, 4181, {1130, 1356, 1695}}, {453, 4681, { 906,  1510, 2265}}, {454, 4994, { 908,  1362, 2724}}, {455, 4103, {1295, 1365, 1443}},
{456, 4106, {1332, 1368, 1406}}, {457, 5027, { 914,  1371, 2742}}, {458, 5038, { 916,  1374, 2748}}, {459, 4135, {1326, 1378, 1431}}, {460, 4163, {1265, 1380, 1518}},
{461, 5071, { 922,  1383, 2766}}, {462, 4168, {1295, 1430, 1443}}, {463, 5093, { 926,  1389, 2778}}, {464, 4188, {1305, 1395, 1488}}, {465, 4206, {1302, 1364, 1540}},
{466, 5126, { 932,  1398, 2796}}, {467, 5137, { 934,  1401, 2802}}, {468, 4213, {1378, 1404, 1431}}, {469, 4757, {1005, 1407, 2345}}, {470, 4358, {1222, 1316, 1820}},
{471, 4867, { 942,  1570, 2355}}, {472, 4297, {1288, 1357, 1652}}, {473, 4307, {1254, 1419, 1634}}, {474, 4660, {1026, 1501, 2133}}, {475, 4315, {1275, 1425, 1615}},
{476, 4301, {1326, 1428, 1547}}, {477, 4334, {1260, 1484, 1590}}, {478, 5258, { 956,  1434, 2868}}, {479, 5269, { 958,  1437, 2874}}, {480, 4323, {1395, 1440, 1488}},
{481, 4331, {1406, 1443, 1482}}, {482, 5302, { 964,  1446, 2892}}, {483, 4353, {1386, 1449, 1518}}, {484, 4444, {1210, 1540, 1694}}, {485, 4559, {1164, 1455, 1940}},
{486, 4464, {1215, 1539, 1710}}, {487, 5357, { 974,  1461, 2922}}, {488, 4440, {1281, 1512, 1647}}, {489, 5053, { 978,  1630, 2445}}, {490, 4423, {1364, 1519, 1540}},
{491, 5401, { 982,  1473, 2946}}, {492, 4482, {1271, 1599, 1612}}, {493, 4492, {1360, 1392, 1740}}, {494, 4554, {1248, 1482, 1824}}, {495, 4463, {1425, 1463, 1575}},
{496, 4513, {1320, 1488, 1705}}, {497, 4817, {1207, 1349, 2261}}, {498, 4782, {1079, 1794, 1909}}, {499, 5489, { 998,  1497, 2994}}, {500, 4525, {1375, 1500, 1650}}}

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作者: ysr    时间: 2026-1-19 14:41


将六分之一拆成两个单位分数的和，不重复结果有哪几种？
https://zhidao.baidu.com/question/492929876883832492.html

设1/6=1/a+1/b（a和b都是大于等于2的正整数）
把式子通分就得到：1/6=(a+b)/ab，进而得到ab=6a+6b, 移项得ab-6a=6b，左边再进行因式分解得a(b-6)=6b…………⑴
或者ab-6b=6a，再进行因式分解得b(a-6)=6a…………⑵
⑴*⑵得：ab(a-6)(b-6)=36ab，约掉ab得(a-6)(b-6)=36
而36=1×36=2×18=3×12=4×9=6×6
因为a和b都是正整数,所以a-6和b-6也都是整数
所以现在有如下可能：（a和b可交换）
1、a-6=1,b-6=36； 解得a=7,b=42
2、a-6=2,b-6=18；解得a=8,b=24
3、a-6=3,b-6=12；解得a=9,b=18
4、a-6=4,b-6=9；解得a=10,b=15
5、a-6=6,b-6=6；解得a=12,b=12
于是六分之一可以写成：
1/6=1/7 + 1/42
1/6=1/8 + 1/24
1/6=1/9 + 1/18
1/6=1/10 + 1/15
1/6=1/12 + 1/12
就这五种可能（网复制的，上面有链接）

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作者: 王守恩    时间: 2026-1-20 15:45
如何把 1/n 拆分成两个不同单位分数的和，并且两个分母之和为最小？——这个就简单了——杀鸡焉用牛刀！

\(a(2)=09,\frac{1}{2}=\frac{1}{3}+\frac{1}{6}\)

\(a(3)=16,\frac{1}{3}=\frac{1}{4}+\frac{1}{12}\)

\(a(4)=18,\frac{1}{4}=\frac{1}{6}+\frac{1}{12}\)

\(a(5)=36,\frac{1}{5}=\frac{1}{6}+\frac{1}{30}\)

\(a(6)=25,\frac{1}{6}=\frac{1}{10}+\frac{1}{15}\)

\(a(7)=64,\frac{1}{7}=\frac{1}{8}+\frac{1}{56}\)

\(a(8)=36,\frac{1}{8}=\frac{1}{12}+\frac{1}{24}\)

\(a(9)=48,\frac{1}{9}=\frac{1}{12}+\frac{1}{36}\)

得到这串数——{9, 16, 18, 36, 25, 64, 36, 48, 45, 144, 49, 196, 63, 64, 72, 324, 75, 400, 81, 100, 99, ——OEIS还没有这串数！

Table[(a + b) /. First@Solve[{1/n == 1/a + 1/b, n (n + 1) >= a > b > n}, {a, b}, Integers], {n, 2, 90}]——通项公式可以简单！

Table[With[{d = Min[Select[Divisors[n^2], # > n &]]}, 2 n + d + n^2/d], {n, 2, 90000}]——或者！！来个高级的！！！！！！

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作者: 王守恩    时间: 2026-1-21 05:37
那串数不是从小到大而是大小波动且不规则，还有重复的，——正确。

通项公式得出来的有时候不是最小值，——不正确——通项公式得出来的肯定是最小值。

拆分成两个单位分数有的是唯一解，如：1/5=1/6+1/30据说就是唯一解，——正确。

{9, 16, 18, 36, 25, 64, 36, 48, 45, 144, 49, 196, 63, 64, 72, 324, 75, 400, 81, 100, 99, 576, 98, 180, 117, 144, 121, 900, 121, 1024, 144, 176, 153, 144, 147, 1444, 171, 208, 162, 1764, 169, 1936, 198, 192, 207, 2304, 196, 448, 225, 272, 234,
2916, 225, 256, 225, 304, 261, 3600, 242, 3844, 279, 256, 288, 324, 275, 4624, 306, 368, 288, 5184, 289, 5476, 333, 320, 342, 324, 325, 6400, 324, 432, 369, 7056, 338, 484, 387, 464, 361, 8100, 361, 400, 414, 496, 423, 576, 392, 9604, 441, 400, 405, 10404, 425, 10816, 441, 432, 477, 11664, 441, 12100, 441, 592, 450, 12996, 475, 784, 522, 484, 531, 576, 484, 1584, 549, 656, 558, 900, 507, 16384, 576, 688, 529, 17424, 529, 676, 603, 576, 612, 19044, 575, 19600, 567, 752, 639, 576, 578, 1044, 657, 700, 666, 22500, 605, 23104, 684, 676, 625, 1116, 625, 24964, 711, 848, 648, 900, 675, 26896, 738, 676, 747, 28224, 675, 2548, 729, 784, 774, 30276, 725, 720, 722, 944, 801, 32400, 722, 33124, 729, 976, 828, 1332, 775, 784, 846, 768, 841, 36864, 784, 37636, 873, 784, 847, 39204, 800, 40000, 810, 1072, 909, 1296, 833, 1476, 927, 1024, 841, 900, 841, 44944, 954, 1136, 963, 1548, 867, 1444, 981, 1168, 882, 900, 925, 50176, 900, 960, 1017, 51984, 931, 52900, 1035, 972, 1044, 54756, 961, 1692, 1062, 1264, 961, 57600, 961, 58564, 1089, 1296, 1098, 1008, 1025, 1024, 1116, 1328, 1125, 63504, 1014, 1156, 1143, 1024, 1152, 66564, 1075, 1936, 1053, 1392, 1179, 69696, 1058, 1908, 1089, 1424, 1206, 72900, 1083, 73984, 1089, 1156, 1233, 1280, 1127, 77284, 1251, 1488, 1125, 79524, 1175, 80656, 1278, 1156, 1152, 2304, 1156, 5508, 1305, 1552, 1314, 86436, 1183, 2124, 1332, 1200, 1341, 1296, 1210, 2500, 1359, 1616, 1225, 2196, 1225, 94864, 1250, 1648, 1395, 97344, 1250, 98596, 1413, 1280, 1422, 101124, 1325, 1600, 1296, 1712, 1369, 1296, 1323,1444, 1467, 1744, 1476, 2916, 1323, 110224, 1494, 1776, 1503, 2412, 1350, 114244, 1521, 1808, 1369, 1764, 1369, 3136, 1548, 1444, 1557, 121104, 1421, 122500, 1440, 1452, 1444, 125316, 1475, 2556, 1602, 1444, 1611, 129600, 1444, 7600, 1629, 1936, 1458, 2628, 1525, 135424, 1521, 1968, 1665, 3392, 1519, 139876, 1521, 1600, 1692, 1764, 1521, 144400, 1521, 2032, 1719, 147456, 1568, 1584, 1737, 2064, 1746, 152100, 1568, 1600, 1575, 2096, 1773, 2844, 1587, 158404, 1791, 1600, 1620, 161604, 1675,1936, 1818, 1728, 1827, 2304, 1666, 168100, 1845, 2192, 1854, 3776, 1681, 2988, 1682, 2224, 1681, 176400, 1681, 178084, 1899, 2256, 1908, 1764, 1775, 3904, 1926, 1728, 1935, 186624, 1734, 188356, 1953, 1856, 1962, 1764, 1825, 193600, 1764, 1792, 1800, 197136, 1813, 3204, 2007, 2384, 1800, 202500, 1805, 2704, 2034, 2416, 2043, 1872, 1849, 209764, 2061, 1936, 1849, 213444, 1849, 215296, 2025, 1984, 2097, 219024, 1875, 4288, 2115, 2512, 2124, 2916, 1975, 1936, 1922, 2544, 2151, 230400, 1922, 2500, 2169, 1936, 2178, 3492, 2025, 238144, 2196, 2608, 2016, 242064, 2009, 2116, 2025, 2000, 2209, 4544, 2075, 250000, 2025, 2672, 2259, 254016, 2023, 3636, 2025, 2704, 2286, 260100, 2048, 4672, 2304, 2116, 2313, 3708, 2107, 3364, 2331, 2768, 2106, 272484, 2175, 274576, 2358, 2116, 2367, 2304, 2116, 13248, 2385, 2832, 2178, 2916, 2225, 3852, 2412, 2864, 2421, 2268, 2166, 293764, 2439, 2896, 2178, 3924, 2187, 300304, 2466, 2928, 2205, 2304, 2209, 5056, 2493, 2368, 2502, 311364, 2325, 3136, 2250, 2352, 2529, 318096, 2303, 4068, 2547, 2304, 2556, 324900, 2299, 327184, 2304, 3056, 2583, 2304, 2312, 334084, 2601, 3088, 2349, 5312, 2425, 4096, 2628, 2352, 2637, 345744, 2366, 2500, 2655, 3152, 2664, 352836, 2400, 2448, 2682, 3184, 2401, 360000, 2401, 362404, 2709, 3216, 2718, 2816, 2525, 369664, 2450, 2500, 2745, 3600, 2450, 376996, 2763, 2624, 2475, 381924, 2575, 384400, 2511, 2500, 2799, 5696, 2500, 4500, 2817, 2700,

---

作者: 王守恩    时间: 2026-1-21 15:43
把 1/n 拆分成两个不同单位分数的和。

\(a(2)=1,\frac{1}{2}=\frac{1}{3}+\frac{1}{6}\)

\(a(3)=1,\frac{1}{3}=\frac{1}{4}+\frac{1}{12}\)

\(a(4)=2,\frac{1}{4}=\frac{1}{6}+\frac{1}{12}=\frac{1}{5}+\frac{1}{20}\)

\(a(5)=1,\frac{1}{5}=\frac{1}{6}+\frac{1}{30}\)

\(a(6)=4,\frac{1}{6}=\frac{1}{10}+\frac{1}{15}=\frac{1}{9}+\frac{1}{18}=\frac{1}{8}+\frac{1}{24}=\frac{1}{7}+\frac{1}{42}\)

\(a(7)=1,\frac{1}{7}=\frac{1}{8}+\frac{1}{56}\)

\(a(8)=3,\frac{1}{8}=\frac{1}{12}+\frac{1}{24}=\frac{1}{10}+\frac{1}{40}=\frac{1}{9}+\frac{1}{72}\)

\(a(9)=2,\frac{1}{9}=\frac{1}{12}+\frac{1}{36}=\frac{1}{10}+\frac{1}{90}\)

解的个数是这样一串数——{1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 13, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 13, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 22, 1, 4, 7, 6, 4, 13, 1, 7, 4, 13, 1, 17, 1, 4, 7, 7, 4, 13, 1, 13, 4, 4, 1, 22, 4, 4, 4, 10, 1, 22, 4, 7, 4, 4, 4, 16, 1, 7, 7, 12, 1, 13, 1, 10, 13, 4, 1, 17, 1, 13, 4, 13, 1, 13, 4, 7, 7, 4, 4, 31, 2, 4, 4, 7, 3, 22, 1, 7, 4, 13, 1, 22, 4, 4, 10, 10, 1, 13, 1, 22, 4, 4, 4, 22, 4, 4, 7, 7, 1, 22, 1, 10, 7, 13, 4, 22, 1, 4, 4, 16, 4, 13, 1, 7, 13, 4, 1, 31, 2, 13, 7, 7, 1, 13, 7, 13, 4, 4, 1, 37, 1, 13, 4, 10, 4, 13, 4, 7, 10, 13, 1, 19, 1, 4, 13, 12, 1, 22, 1, 17, 4, 4, 4, 22, 4, 4, 7, 13, 4, 40, 1, 7, 4, 4, 4, 24, 4, 4, 4, 22, 4, 13, 1, 16, 12, 4, 1, 22, 1, 13, 13, 10, 1, 22, 4, 7, 4, 13, 1, 40, 1, 7, 5, 7, 7, 13, 4, 10, 4, 10, 1, 37, 4, 4, 13, 8, 1, 13, 4, 22, 7, 4, 1, 31, 4, 13, 4, 7, 1, 31, 1, 13, 13, 4, 7, 22, 1, 4, 7, 31, 1, 13, 1, 7, 13, 13, 4, 27, 2, 13, 4, 7, 1, 22, 4, 10, 10, 4, 4, 37, 4, 4, 4, 13, 4, 22, 1, 22, 4, 13, 1, 31, 1, 4, 22, 7, 1, 13, 4, 19, 4, 13, 4, 22, 7, 4, 4, 10, 4, 40, 1, 7, 7, 4, 4, 40, 1, 7, 4, 22, 4, 22, 3, 10, 13, 4, 1, 22, 1, 22, 10, 16, 1, 13, 4, 7, 13, 4, 1, 52, 2, 4, 7, 22, 4, 13, 1, 13, 7, 13, 4, 22, 1, 13, 10, 10, 4, 31, 1, 22, 4, 4, 1, 22, 13, 4, 7, 7, 1, 40, 4, 17, 4, 4, 4, 37, 1, 4, 13, 22, 1, 13, 4, 7, 13, 13, 4, 31, 1, 13, 4, 7, 4, 22, 4, 16, 4, 13, 1, 67, 1, 4, 7, 10, 7, 13, 4, 7, 13, 13, 1, 31, 1, 13, 13, 7, 4, 13, 1, 31, 12, 13, 1, 22, 4, 4, 4, 19, 1, 37, 4, 7, 4, 4, 13, 31, 1, 4, 10, 22, 1, 40, 1, 13, 13, 4, 1, 37, 4, 13, 4, 10, 4, 13, 7, 22, 7, 4, 1, 49, 4, 4, 13, 12, 4, 16, 1, 10, 4, 22, 1, 22, 4, 13, 22, 13, 4, 13, 1, 17, 4, 4, 1, 52, 4, 13, 7, 7, 1, 40, 4, 9, 10, 4, 4, 22, 4, 13, 4, 31, 1, 22, 1, 7, 22, 4, 4, 40, 2, 13, 7, 22, 4, 13, 4, 10, 4, 4, 7, 52, 1, 4, 4, 16, 4, 40, 1, 7, 7, 22, 4, 31, 4, 4, 13, 7, 1, 22, 4, 40, 13, 4, 1, 22, 4, 4, 13, 10, 1, 40, 1, 22, 4, 13, 7, 32, 1, 7, 4, 22, 4, 13, 4, 10, 22, 4, 1, 37, 4, 13, 4, 13, 1, 31, 13, 7, 4, 13, 1, 52, 1, 13, 7, 7, 7, 13, 1, 16, 13, 13, 4, 37, 1, 4, 13, 31, 1, 13, 1, 22, 10, 4, 4, 40, 4, 4, 13, 7, 4, 67, 1, 10, 4, 4, 4, 22, 7, 13, 7, 22, 1, 13, 1, 22, 13, 13, 1, 31, 4, 22, 13, 7, 1, 13, 4, 13, 7, 13, 1, 67, 1, 4, 13, 10, 13, 22, 4, 7, 4, 13, 4, 49, 1, 4, 17, 12, 1, 13, 4, 31, 4, 13, 1, 37, 4, 10, 4, 13, 4, 40, 1, 7, 22, 4, 4, 31, 4, 4, 4, 37, 1, 31, 4, 19, 13, 4, 4, 22, 1, 13, 7, 10, 4, 40, 13, 7, 4, 4, 1, 67, 4, 7, 4, 7, 7, 22, 1, 31, 6, 13, 4, 22, 1, 4, 22, 16, 4, 22, 1, 22, 13, 13, 1, 31, 4, 4, 7, 22, 4, 31, 1, 13, 4, 13, 4, 52, 1, 4, 13, 31, 1, 13, 4, 7, 22, 4, 4, 25, 1, 40, 4, 7, 1, 22, 7, 10, 13, 4, 4, 67, 4, 13, 10, 22, 4, 13, 1, 7, 4, 13, 4, 52, 4, 4, 13, 7, 1, 40, 4, 27, 7, 4, 4, 22, 13, 13, 4, 10, 1, 40, 1, 22, 4, 13, 4, 40, 4, 4, 22, 22, 1, 13, 1, 10, 22, 13, 1, 37, 1, 13, 4, 19, 7, 13, 4, 22, 10, 4, 1, 94, 2, 4, 4, 7, 7, 22, 7, 13, 4, 22, 4, 22, 1, 13, 22, 10, 1, 40, 1, 22, 13, 4, 1, 38, 4, 4, 7, 22, 4, 40, 4, 10, 7, 13, 10, 22, 1, 4, 4, 40, 1, 37, 1, 22, 13, 4, 1, 31, 4, 13, 13, 7, 4, 13, 4, 22, 13, 4, 4, 62}
Table[(Length[Divisors[n^2]] - 1)/2, {n, 2, 900}]——这是通项公式。

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作者: ysr    时间: 2026-1-21 16:10

王守恩 发表于 2026-1-21 07:43
把 1/n 拆分成两个不同单位分数的和。

\(a(2)=1,\frac{1}{2}=\frac{1}{3}+\frac{1}{6}\)

6  25  {10,15}
需要比较的解的个数为： 4组，额，是从2开始的，6是四组，去掉了一组两个分母相同的就是1/6=1/12+1/12

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作者: ysr    时间: 2026-1-21 18:19
程序代码如下：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = 2 * (n + 1) ^ 2
a = Val(n + 1)
Do While a <= Val(2 * n)
b = Val(2 * n)
Do While b <= Val(n * (n + 1)) And b > a
k = Val(a + b)
c = Val(a * b) / Val(k)
If c = n And k <= m Then
m = k
s = "{" & a & "," & b & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If

b = Val(b + 1)
Loop
a = Val(a + 1)
Loop

If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If

End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""

End Sub

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