1/n=1/a+1/b+1/c+1/d

王守恩

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？

\(a(1)=24,\frac{1}{1}=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{12}\)     
\(a(2)=43,\frac{1}{2}=\frac{1}{5}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}\)   
\(a(3)=52,\frac{1}{3}=\frac{1}{9}+\frac{1}{10}+\frac{1}{15}+\frac{1}{18}\)      
\(a(4)=74,\frac{1}{4}=\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+\frac{1}{28}\)      
\(a(5)=84,\frac{1}{5}=\frac{1}{15}+\frac{1}{20}+\frac{1}{21}+\frac{1}{28}\)
\(a(6)=99,\frac{1}{6}=\frac{1}{20}+\frac{1}{21}+\frac{1}{28}+\frac{1}{30}\)
\(a(7)=120,\frac{1}{7}=\frac{1}{21}+\frac{1}{24}+\frac{1}{35}+\frac{1}{40}\)   
\(a(8)=135,\frac{1}{8}=\frac{1}{24}+\frac{1}{30}+\frac{1}{36}+\frac{1}{45}\)   
\(a(9)=153,\frac{1}{9}=\frac{1}{26}+\frac{1}{36}+\frac{1}{39}+\frac{1}{52}\)

—答案是个错家伙？——OEIS没有这串数？

northwolves

{1,{2,4,6,12},24}
{2,{4,10,12,15},41}
{3,{9,10,15,18},52}
{4,{10,15,21,28},74}
{5,{15,20,21,28},84}
{6,{20,21,28,30},99}
{7,{20,28,30,42},120}
{8,{24,30,36,45},135}
{9,{26,36,39,52},153}
{10,{30,40,42,56},168}
{11,{36,44,45,55},180}
{12,{40,42,56,60},198}
{13,{39,52,60,65},216}
{14,{45,55,63,66},229}
{15,{50,55,66,75},246}
{16,{55,60,66,80},261}

王守恩

northwolves 发表于 2026-1-16 15:49
{1,{2,4,6,12},24}
{2,{4,10,12,15},41}
{3,{9,10,15,18},52}

这代码有问题吗？——心里没底气。谢谢！

1. 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}}}

王守恩

同理。这代码有问题吗？——心里没底气。谢谢！
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;
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}]——这是3#的代码。

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++, 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; 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}]

{{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

把基础资料补上——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).

03, 011=2+3+6,
04, 024=2+4+6+12,
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,

后面的没有了？后面的就没有了吗？！

王守恩

—答案没错了！——OEIS还是找不到这串数！

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

太难！丢了。换一道。

\(a(3)=11=6+3+2,\frac{1}{6}+\frac{1}{3}+\frac{1}{2}\)

\(a(4)=12=6+3+1+2,\frac{1}{6}+\frac{1}{3}+1-\frac{1}{2}\)

\(a(5)=22=12+2+1+3+4,\frac{1}{12}+\frac{1}{2}+1-\frac{1}{3}-\frac{1}{4}\)

\(a(6)=28=12+4+3+1+2+6,\frac{1}{12}+\frac{1}{4}+\frac{1}{3}+1-\frac{1}{2}-\frac{1}{6}\)

\(a(7)=49=20+12+5+3+1+2+6,\frac{1}{20}+\frac{1}{12}+\frac{1}{5}+\frac{1}{3}+1-\frac{1}{2}-\frac{1}{6}\)

\(a(8)=52=15+5+2+1+3+4+10+12,\frac{1}{15}+\frac{1}{5}+\frac{1}{2}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{10}-\frac{1}{12}\)

\(a(9)=73=30+10+5+2+1+3+4+6+12,\frac{1}{30}+\frac{1}{10}+\frac{1}{5}+\frac{1}{2}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{6}-\frac{1}{12}\)

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？

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

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

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

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

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

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

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

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

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}]——通项公式也是太简单了！！！

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

magicstrawberry

王守恩 发表于 2026-1-20 10:39
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？

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

设n^2分解为n^2=b*c，要求b≠c且|b-c|最小。这样的b+c+2n即为所求答案。

王守恩

7#整理一下。——蛮好玩的。——好像有规律——手工还是困难。

\(a(3)=11=2+3+6,\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1\)

\(a(4)=12=2+1+3+6,\frac{1}{2}+1-\frac{1}{3}-\frac{1}{6}=1\)

\(a(5)=22=2+12+1+3+4,\frac{1}{2}+\frac{1}{12}+1-\frac{1}{3}-\frac{1}{4}=1\)

\(a(6)=28=2+6+1+3+4+12,\frac{1}{2}+\frac{1}{6}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{12}=1\)

\(a(7)=41=2+4+1+3+5+6+20,\frac{1}{2}+\frac{1}{4}+1-\frac{1}{3}-\frac{1}{5}-\frac{1}{6}-\frac{1}{20}=1\)

\(a(8)=51=2+4+20+1+3+5+6+10,\frac{1}{2}+\frac{1}{4}+\frac{1}{20}+1-\frac{1}{3}-\frac{1}{5}-\frac{1}{6}-\frac{1}{10}=1\)

\(a(9)=58=2+4+10+1+3+5+6+12+15,\frac{1}{2}+\frac{1}{4}+\frac{1}{10}+1-\frac{1}{3}-\frac{1}{5}-\frac{1}{6}-\frac{1}{12}-\frac{1}{15}=1\)

\(a(10)=78=2+6+8+10+1+3+4+5+15+24,\frac{1}{2}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{15}-\frac{1}{24}=1\)

\(a(11)=90=2+6+8+10+24+1+3+4+5+12+15,\frac{1}{2}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{24}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{12}-\frac{1}{15}=1\)

\(a(12)=100=2+6+7+10+14+1+3+4+5+12+15+21,\frac{1}{2}+\frac{1}{6}+\frac{1}{7}+\frac{1}{10}+\frac{1}{14}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{12}-\frac{1}{15}-\frac{1}{21}=1\)

\(a(13)=120=2+6+7+10+14+24+1+3+4+5+8+15+21,\frac{1}{2}+\frac{1}{6}+\frac{1}{7}+\frac{1}{10}+\frac{1}{14}+\frac{1}{24}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{8}-\frac{1}{15}-\frac{1}{21}=1\)

\(a(14)=132=2+6+7+10+12+14+1+3+4+5+8+15+21+24,\frac{1}{2}+\frac{1}{6}+\frac{1}{7}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{8}-\frac{1}{15}-\frac{1}{21}-\frac{1}{24}=1\)

\(a(15)=165=2+6+7+9+10+14+1+3+4+5+8+15+21+24+36,\frac{1}{2}+\frac{1}{6}+\frac{1}{7}+\frac{1}{9}+\frac{1}{10}+\frac{1}{14}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{8}-\frac{1}{15}-\frac{1}{21}-\frac{1}{24}-\frac{1}{36}=1\)