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？

天山草

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？

天山草

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。

天山草

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

tmduser

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

天山草

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？

王守恩

{{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

王守恩

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

ysr

https://www.toutiao.com/article/ ... /?wid=1768555423251

ysr

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

ysr

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