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

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？！