1/n=1/a+1/b+1/c+1/d
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？ {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} 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！
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 == 0, d = u/v; If]]]]]; {n, M, S}], {n, 100}]
{{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}}} 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 == 0, d = u/v; If]]]]]; {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 == 0, f = u/v; If]]]]]];{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}}} 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？！ —答案没错了！——OEIS还是找不到这串数！
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;
v = a*b*c - n (a*b + (a + b) c); If == 0, d = u/v; If]]]]]; {n, M, S}], {n, 200}]
{{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}}} 太难！丢了。换一道。
\(a(3)=11=6+3+2,\frac{1}{6}+\frac{1}{3}+\frac{1}{2}\)
\(a(4)=12=6+3+1+2,\frac{1}{6}+\frac{1}{3}+1-\frac{1}{2}\)
\(a(5)=22=12+2+1+3+4,\frac{1}{12}+\frac{1}{2}+1-\frac{1}{3}-\frac{1}{4}\)
\(a(6)=28=12+4+3+1+2+6,\frac{1}{12}+\frac{1}{4}+\frac{1}{3}+1-\frac{1}{2}-\frac{1}{6}\)
\(a(7)=49=20+12+5+3+1+2+6,\frac{1}{20}+\frac{1}{12}+\frac{1}{5}+\frac{1}{3}+1-\frac{1}{2}-\frac{1}{6}\)
\(a(8)=52=15+5+2+1+3+4+10+12,\frac{1}{15}+\frac{1}{5}+\frac{1}{2}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{10}-\frac{1}{12}\)
\(a(9)=73=30+10+5+2+1+3+4+6+12,\frac{1}{30}+\frac{1}{10}+\frac{1}{5}+\frac{1}{2}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{6}-\frac{1}{12}\) 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, # > n &]]}, 2 n + d + n^2/d], {n, 2, 90000}]——来个高级的！！！！！！ 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。 7#整理一下。——蛮好玩的。——好像有规律——手工还是困难。
\(a(3)=11=2+3+6,\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1\)
\(a(4)=12=2+1+3+6,\frac{1}{2}+1-\frac{1}{3}-\frac{1}{6}=1\)
\(a(5)=22=2+12+1+3+4,\frac{1}{2}+\frac{1}{12}+1-\frac{1}{3}-\frac{1}{4}=1\)
\(a(6)=28=2+6+1+3+4+12,\frac{1}{2}+\frac{1}{6}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{12}=1\)
\(a(7)=41=2+4+1+3+5+6+20,\frac{1}{2}+\frac{1}{4}+1-\frac{1}{3}-\frac{1}{5}-\frac{1}{6}-\frac{1}{20}=1\)
\(a(8)=51=2+4+20+1+3+5+6+10,\frac{1}{2}+\frac{1}{4}+\frac{1}{20}+1-\frac{1}{3}-\frac{1}{5}-\frac{1}{6}-\frac{1}{10}=1\)
\(a(9)=58=2+4+10+1+3+5+6+12+15,\frac{1}{2}+\frac{1}{4}+\frac{1}{10}+1-\frac{1}{3}-\frac{1}{5}-\frac{1}{6}-\frac{1}{12}-\frac{1}{15}=1\)
\(a(10)=78=2+6+8+10+1+3+4+5+15+24,\frac{1}{2}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{15}-\frac{1}{24}=1\)
\(a(11)=90=2+6+8+10+24+1+3+4+5+12+15,\frac{1}{2}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{24}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{12}-\frac{1}{15}=1\)
\(a(12)=100=2+6+7+10+14+1+3+4+5+12+15+21,\frac{1}{2}+\frac{1}{6}+\frac{1}{7}+\frac{1}{10}+\frac{1}{14}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{12}-\frac{1}{15}-\frac{1}{21}=1\)
\(a(13)=120=2+6+7+10+14+24+1+3+4+5+8+15+21,\frac{1}{2}+\frac{1}{6}+\frac{1}{7}+\frac{1}{10}+\frac{1}{14}+\frac{1}{24}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{8}-\frac{1}{15}-\frac{1}{21}=1\)
\(a(14)=132=2+6+7+10+12+14+1+3+4+5+8+15+21+24,\frac{1}{2}+\frac{1}{6}+\frac{1}{7}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{8}-\frac{1}{15}-\frac{1}{21}-\frac{1}{24}=1\)
\(a(15)=165=2+6+7+9+10+14+1+3+4+5+8+15+21+24+36,\frac{1}{2}+\frac{1}{6}+\frac{1}{7}+\frac{1}{9}+\frac{1}{10}+\frac{1}{14}+1-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{8}-\frac{1}{15}-\frac{1}{21}-\frac{1}{24}-\frac{1}{36}=1\)