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？

ysr

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！

ysr

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！

王守恩

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,

王守恩

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}}}

王守恩

—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}}}

王守恩

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}}}

ysr

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）

王守恩

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！！！！！！

王守恩

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,

王守恩

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。