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正整数A为A数字乘积的最小数

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发表于 2026-10-28 14:48 | 显示全部楼层 |阅读模式
正整数A为A数字乘积的最小数。——来个通项公式。

{1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 35, 37, 38, 39, 45, 47, 48, 49, 55, 56, 57, 58, 59, 67, 68, 69, 77, 78, 79, 88, 89, 99, 255, 256,257, 258, 259, 267, 268, 269, 277, 278,279, 288, 289,299, 355, 357, 358,
359, 377, 378, 379, 388, 389, 399, 455, 457, 458, 459, 477, 478,479, 488, 489, 499, 555, 556, 557, 558, 559, 567, 568, 569, 577,578, 579, 588,589, 599, 677, 678,679, 688, 689, 699, 777,778, 779, 788, 789,
799, 888, 889, 899, 999, 2555, 2556, 2557, 2558, 2559, 2567, 2568, 2569, 2577, 2578, 2579, 2588, 2589, 2599, 2677, 2678, 2679, 2688, 2689, 2699, 2777, 2778, 2779, 2788, 2789, 2799, 2888, 2889, 2899,
2999, 3555, 3557, 3558, 3559, 3577, 3578, 3579, 3588, 3589, 3599, 3777, 3778, 3779, 3788, 3789, 3799, 3888, 3889, 3899, 3999, 4555, 4557, 4558, 4559, 4577, 4578, 4579, 4588, 4589, 4599, 4777, 4778,
4779, 4788, 4789, 4799, 4888, 4889, 4899, 4999, 5555, 5556, 5557, 5558, 5559, 5567, 5568, 5569, 5577, 5578, 5579, 5588, 5589, 5599, 5677, 5678, 5679, 5688, 5689, 5699, 5777, 5778, 5779, 5788, 5789,
5799, 5888, 5889, 5899, 5999, 6777, 6778, 6779, 6788, 6789, 6799, 6888, 6889, 6899, 6999, 7777, 7778, 7779, 7788, 7789, 7799, 7888, 7889, 7899, 7999, 8888, 8889, 8899, 8999, 9999, 25555, 25556,

maxM = 10^7; res = ConstantArray[0, maxM + 1]; gen[len_, minD_, maxD_] := gen[len, minD, maxD] = If[len == 0, {{}}, Flatten[Table[Prepend[#, d] & /@ gen[len - 1, d, maxD], {d, minD, maxD}], 1]];
Do[n = FromDigits[seq]; m = Times @@ seq; If[m <= maxM && (res[[m + 1]] == 0 || n < res[[m + 1]]), res[[m + 1]] = n], {len, 1, 11}, {seq, gen[len, 1, 9]}]; Union@Select[res, 0 < # <= maxM &]
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发表于 2026-10-28 22:01 | 显示全部楼层
都是十进制惹的祸。
人类采用11进制就是另一个明月。
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 楼主| 发表于 2026-10-29 07:14 | 显示全部楼层
A 是 A 数字乘积的最小数。——这串数肯定有漂亮的公式！！！——不要期望AI,  AI是搞不了的！

{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}——"1"只出现1次——抛！

{25, 26, 27, 28, 29}——2的后面不会是2, 3, 4。因为2×2=4,  2×3=6,  2×4=8。
{55, 56, 57, 58, 59}——5的后面不会是2, 3, 4。
{35, 37, 38, 39}——3的后面不会是3, 4, 6。因为3×3=9,  3×4=2×6,  3×6=2×9。
{45, 47, 48, 49}——4的后面不会是3, 4, 6。因为4×4=2×8,  4×6=3×8。
{67, 68, 69}——6的后面不会是6。因为6×6=4×9。
{77, 78, 79}——7的后面不会是6。
{88, 89}——2个。
{99}——最大的2位数。

{255, 256, 257, 258, 259, 267, 268, 269, 277, 278, 279, 288, 289, 299}——"2"与"5"尾数相同。
{555, 556, 557, 558, 559, 567, 568, 569, 577, 578, 579, 588, 589, 599},
{355, 357, 358, 359, 377, 378, 379, 388, 389, 399}——"3"与"4"尾数相同。
{455, 457, 458, 459, 477, 478, 479, 488, 489, 499},
{677, 678, 679, 688, 689, 699}——"6"与"7"尾数相同。
{777, 778, 779, 788, 789, 799},
{888, 889, 899}——3个。
{999}——最大的3位数。

{2555, 2556, 2557, 2558, 2559, 2567, 2568, 2569, 2577, 2578, 2579, 2588, 2589, 2599, 2677, 2678, 2679, 2688, 2689, 2699, 2777, 2778, 2779, 2788, 2789, 2799, 2888, 2889, 2899, 2999},
{5555, 5556, 5557, 5558, 5559, 5567, 5568, 5569, 5577, 5578, 5579, 5588, 5589, 5599, 5677, 5678, 5679, 5688, 5689, 5699, 5777, 5778, 5779, 5788, 5789, 5799, 5888, 5889, 5899, 5999},
{3555, 3557, 3558, 3559, 3577, 3578, 3579, 3588, 3589, 3599, 3777, 3778, 3779, 3788, 3789, 3799, 3888, 3889, 3899, 3999},
{4555, 4557, 4558, 4559, 4577, 4578, 4579, 4588, 4589, 4599, 4777, 4778, 4779, 4788, 4789, 4799, 4888, 4889, 4899, 4999},
{6777, 6778, 6779, 6788, 6789, 6799, 6888, 6889, 6899, 6999},
{7777, 7778, 7779, 7788, 7789, 7799, 7888, 7889, 7899, 7999},
{8888, 8889, 8899, 8999}——4个。
{9999}——最大的4位数。

{25555, 25556, 25557, 25558, 25559, 25567, 25568, 25569, 25577, 25578, 25579, 25588, 25589, 25599, 25677, 25678, 25679, 25688, 25689, 25699, 25777, 25778, 25779, 25788, 25789, 25799, 25888, 25889, 25899, 25999,
26777, 26778, 26779, 26788, 26789, 26799, 26888, 26889, 26899, 26999, 27777, 27778, 27779, 27788, 27789, 27799, 27888, 27889, 27899, 27999, 28888, 28889, 28899, 28999, 29999},
{55555, 55556, 55557, 55558, 55559, 55567, 55568, 55569, 55577, 55578, 55579, 55588, 55589, 55599, 55677, 55678, 55679, 55688, 55689, 55699, 55777, 55778, 55779, 55788, 55789, 55799, 55888, 55889, 55899, 55999,
56777, 56778, 56779, 56788, 56789, 56799, 56888, 56889, 56899, 56999, 57777, 57778, 57779, 57788, 57789, 57799, 57888, 57889, 57899, 57999, 58888, 58889, 58899, 58999, 59999},
{35555, 35557, 35558, 35559, 35577, 35578, 35579, 35588, 35589, 35599, 35777, 35778, 35779, 35788, 35789, 35799, 35888, 35889, 35899, 35999,
37777, 37778, 37779, 37788, 37789, 37799, 37888, 37889, 37899, 37999, 38888, 38889, 38899, 38999, 39999},
{45555, 45557, 45558, 45559, 45577, 45578, 45579, 45588, 45589, 45599, 45777, 45778, 45779, 45788, 45789, 45799, 45888, 45889, 45899, 45999,
47777, 47778, 47779, 47788, 47789, 47799, 47888, 47889, 47899, 47999, 48888, 48889, 48899, 48999, 49999},
{67777, 67778, 67779, 67788, 67789, 67799, 67888, 67889, 67899, 67999, 68888, 68889, 68899, 68999, 69999},
{77777, 77778, 77779, 77788, 77789, 77799, 77888, 77889, 77899, 77999, 78888, 78889, 78899, 78999, 79999},
{88888, 88889, 88899, 88999, 89999}——5个。
{99999}——最大的5位数。
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 楼主| 发表于 2026-10-29 08:48 | 显示全部楼层
A 是 A 数字乘积的最小数。

1位数有9个。
9=1(数码2)+1(数码5)+1(数码3)+1(数码4)+1(数码6)+1(数码7)+1(数码8)+1(数码9)+1(数码1)。

2位数有27个。1,2位数有36个。
27=5(数码2)+5(数码5)+4(数码3)+4(数码4)+3(数码6)+3(数码7)+2(数码8)+1(数码9)。

3位数有64个。1,2,3位数有100个。
64=14(数码2)+14(数码5)+10(数码3)+10(数码4)+6(数码6)+6(数码7)+3(数码8)+1(数码9)。

4位数有125个。1,2,3,4位数有225个。
125=30(数码2)+30(数码5)+20(数码3)+20(数码4)+10(数码6)+10(数码7)+4(数码8)+1(数码9)。

5位数有216个。1,23,4,5位数有441个。
216=55(数码2)+55(数码5)+35(数码3)+35(数码4)+15(数码6)+15(数码7)+5(数码8)+1(数码9)。

6位数有343个。1,2,3,4,5,6位数有784个。
343=91(数码2)+91(数码5)+56(数码3)+56(数码4)+21(数码6)+21(数码7)+6(数码8)+1(数码9)。
......

{9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, 11025, 14400, 18496, 23409, 29241, 36100, 44100, 53361, 64009, 76176,90000,105625,123201,——A000537

{9, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824,  15625, 17576, 19683, 21952,——A000578

{1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930,7714,8555,9455,10416,—— A000330

{1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984,——A000292

{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741,——A000217

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,  37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,——A000027

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,——A000012
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 楼主| 发表于 2026-10-29 12:02 | 显示全部楼层
A 是 A 数字乘积的最小数。——OEIS有这串数。

A343403——Numbers k such that the product of the digits of k is not the product of digits of any earlier term in the sequence.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 35, 37, 38, 39, 45, 47, 48, 49, 55, 56, 57, 58, 59, 67, 68, 69, 77, 78, 79, 88, 89, 99, 255, 256, 257, 258, 259, 267, 268, 269, 277, 278, 279, 288, 289, 299, 355, 357, 358, 359, 377, 378,

我们有伟大的公式！——可以到无穷——给出前23位数。
  1. V = <|2 -> Range[5, 9], 5 -> Range[5, 9], 3 -> {5, 7, 8, 9}, 4 -> {5, 7, 8, 9}, 6 -> Range[7, 9], 7 -> Range[7, 9], 8 -> Range[8, 9], 9 -> {9}|>;
  2. W = <||>; F[n_] := Module[{m = Times @@ IntegerDigits[n]}, If[! KeyExistsQ[W, m] || n < W[m], W[m] = n]]; F[1];
  3. A[T_, S_] := Flatten[Table[x*10 + d, {x, T}, {d, Select[S, # >= Last[IntegerDigits[x]] &]}]];
  4. Do[B = Flatten[NestList[A[#, V[d]] &, {d}, 22]]; F /@ B, {d, {2, 5, 3, 4, 6, 7, 8, 9}}];
  5. Take[Sort[Values[W]], 300^2]
复制代码

\!\(
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     "7", ",", "8", ",", "9", ",", "25", ",", "26", ",", "27", ",",
     "28", ",", "29", ",", "35", ",", "37", ",", "38", ",", "39", ",",
      "45", ",", "47", ",", "48", ",", "49", ",", "55", ",", "56",
     ",", "57", ",", "58", ",", "59", ",", "67", ",", "68", ",", "69",
      ",", "77", ",", "78", ",", "79", ",", "88", ",", "89", ",",
TemplateBox[{"89930"},
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     "78889999999999999999999", ",", "78899999999999999999999", ",",
     "78999999999999999999999", ",", "79999999999999999999999", ",",
     "88888888888888888888888", ",", "88888888888888888888889", ",",
     "88888888888888888888899", ",", "88888888888888888888999", ",",
     "88888888888888888889999", ",", "88888888888888888899999", ",",
     "88888888888888888999999", ",", "88888888888888889999999", ",",
     "88888888888888899999999", ",", "88888888888888999999999", ",",
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     "88888888888999999999999", ",", "88888888889999999999999", ",",
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     "88888889999999999999999", ",", "88888899999999999999999", ",",
     "88888999999999999999999", ",", "88889999999999999999999", ",",
     "88899999999999999999999", ",", "88999999999999999999999", ",",
     "89999999999999999999999", ",", "99999999999999999999999"}],
    "}"}],
Short[#, 5]& ]\)
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 楼主| 发表于 2026-10-29 14:55 | 显示全部楼层
伟大的公式！——可以到无穷——给出前 1—99 位数。合计 9(1位数)+25502482(中间项)+9(99位的最后9项)=2552500项。
{1, 2, 3, 4, 5, 6, 7, 8, 9, OutputSizeLimit`Skeleton[25502482],
888888889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,
888888899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,
888888999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,
888889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,
888899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,
888999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,
889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,
899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,
999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999}
  1. V = <|2 -> Range[5, 9], 5 -> Range[5, 9], 3 -> {5, 7, 8, 9}, 4 -> {5, 7, 8, 9}, 6 -> Range[7, 9], 7 -> Range[7, 9], 8 -> Range[8, 9], 9 -> {9}|>;
  2. W = <||>; F[n_] := Module[{m = Times @@ IntegerDigits[n]}, If[! KeyExistsQ[W, m] || n < W[m], W[m] = n]]; F[1];
  3. A[T_, S_] := Flatten[Table[x*10 + d, {x, T}, {d, Select[S, # >= Last[IntegerDigits[x]] &]}]];
  4. Do[B = Flatten[NestList[A[#, V[d]] &, {d}, 99]]; F /@ B, {d, {2, 5, 3, 4, 6, 7, 8, 9}}];
  5. Take[Sort[Values[W]], 5050^2]
复制代码
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 楼主| 发表于 2026-10-30 08:21 | 显示全部楼层
这个代码完全体现了 3#, 4# 的思想。——速度比 A343403快多了！
  1. U = 5; V = <|2 -> Range[5, 9], 5 -> Range[5, 9], 3 -> {5, 7, 8, 9}, 4 -> {5, 7, 8, 9}, 6 -> {7, 8, 9}, 7 -> {7, 8, 9}, 8 -> {8, 9}, 9 -> {9}|>;
  2. g[s_, t_] := Flatten[Table[10 x + d, {x, s}, {d, Select[t, # >= Last[IntegerDigits[x]] &]}]];
  3. Take[DeleteDuplicatesBy[Sort[Flatten[Join[{{1}}, Table[NestList[g[#, V[d]] &, {d}, U], {d, 2, 9}]], 2]], Times @@ IntegerDigits[#] &], ((U + 2) (U + 3)/2)^2]
复制代码

{1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 35, 37, 38, 39, 45, 47, 48, 49, 55, 56, 57, 58, 59, 67, 68, 69, 77, 78, 79, 88, 89, 99,
255, 256, 257, 258, 259, 267, 268, 269, 277, 278, 279, 288, 289, 299, 355, 357, 358, 359, 377, 378, 379, 388, 389, 399, 455, 457, 458, 459, 477, 478, 479, 488, 489, 499,
555, 556, 557, 558, 559, 567, 568, 569, 577, 578, 579, 588, 589, 599, 677, 678, 679, 688, 689, 699, 777, 778, 779, 788, 789, 799, 888, 889, 899, 999,
2555, 2556, 2557, 2558, 2559, 2567, 2568, 2569, 2577, 2578, 2579, 2588, 2589, 2599, 2677, 2678, 2679, 2688, 2689, 2699, 2777, 2778, 2779, 2788, 2789,2799, 2888, 2889, 2899, 2999, 3555, 3557,
3558, 3559, 3577, 3578, 3579, 3588, 3589, 3599, 3777, 3778, 3779, 3788, 3789, 3799, 3888, 3889, 3899, 3999, 4555, 4557, 4558, 4559, 4577, 4578, 4579, 4588, 4589, 4599, 4777, 4778, 4779, 4788,
4789, 4799, 4888, 4889, 4899, 4999, 5555, 5556, 5557, 5558, 5559, 5567, 5568, 5569, 5577, 5578, 5579, 5588, 5589, 5599, 5677, 5678, 5679, 5688, 5689, 5699, 5777, 5778, 5779, 5788, 5789, 5799,
5888, 5889, 5899, 5999, 6777, 6778, 6779, 6788, 6789, 6799, 6888, 6889, 6899, 6999, 7777, 7778, 7779, 7788, 7789, 7799, 7888, 7889, 7899, 7999, 8888, 8889, 8899, 8999, 9999,
25555, 25556, 25557, 25558, 25559, 25567, 25568, 25569, 25577, 25578, 25579, 25588, 25589, 25599, 25677, 25678, 25679, 25688, 25689, 25699, 25777, 25778, 25779, 25788, 25789, 25799, 25888, 25889, 25899, 25999, 26777,
26778, 26779, 26788, 26789, 26799, 26888, 26889, 26899, 26999, 27777, 27778, 27779, 27788, 27789, 27799, 27888, 27889, 27899, 27999, 28888, 28889, 28899, 28999, 29999, 35555, 35557, 35558, 35559, 35577, 35578, 35579,
35588, 35589, 35599, 35777, 35778, 35779, 35788, 35789, 35799, 35888, 35889, 35899, 35999, 37777, 37778, 37779, 37788, 37789, 37799, 37888, 37889, 37899, 37999, 38888, 38889, 38899, 38999, 39999, 45555, 45557, 45558,
45559, 45577, 45578, 45579, 45588, 45589, 45599, 45777, 45778, 45779, 45788, 45789, 45799, 45888, 45889, 45899, 45999, 47777, 47778, 47779, 47788, 47789, 47799, 47888, 47889, 47899, 47999, 48888, 48889, 48899, 48999,
49999, 55555, 55556, 55557, 55558, 55559, 55567, 55568, 55569, 55577, 55578, 55579, 55588, 55589, 55599, 55677, 55678, 55679, 55688, 55689, 55699, 55777, 55778, 55779, 55788, 55789, 55799, 55888, 55889, 55899, 55999,
56777, 56778, 56779, 56788, 56789, 56799, 56888, 56889, 56899, 56999, 57777, 57778, 57779, 57788, 57789, 57799, 57888, 57889, 57899, 57999, 58888, 58889, 58899, 58999, 59999, 67777, 67778, 67779, 67788, 67789, 67799,
67888, 67889, 67899, 67999, 68888, 68889, 68899, 68999, 69999, 77777, 77778, 77779, 77788, 77789, 77799, 77888, 77889, 77899, 77999, 78888, 78889, 78899, 78999, 79999, 88888, 88889, 88899, 88999, 89999, 99999,
255555, 255556, 255557, 255558, 255559, 255567, 255568, 255569, 255577, 255578, 255579, 255588, 255589, 255599, 255677, 255678, 255679, 255688, 255689, 255699, 255777, 255778, 255779, 255788, 255789, 255799, 255888,
255889, 255899, 255999, 256777, 256778, 256779, 256788, 256789, 256799, 256888, 256889, 256899, 256999, 257777, 257778, 257779, 257788, 257789, 257799, 257888, 257889, 257899, 257999, 258888, 258889, 258899, 258999,
259999, 267777, 267778, 267779, 267788, 267789, 267799, 267888, 267889, 267899, 267999, 268888, 268889, 268899, 268999, 269999, 277777, 277778, 277779, 277788, 277789, 277799, 277888, 277889, 277899, 277999, 278888,
278889, 278899, 278999, 279999, 288888, 288889, 288899, 288999, 289999, 299999, 355555, 355557, 355558, 355559, 355577, 355578, 355579, 355588, 355589, 355599, 355777, 355778, 355779, 355788, 355789, 355799, 355888,
355889, 355899, 355999, 357777, 357778, 357779, 357788, 357789, 357799, 357888, 357889, 357899, 357999, 358888, 358889, 358899, 358999, 359999, 377777, 377778, 377779, 377788, 377789, 377799, 377888, 377889, 377899,
377999, 378888, 378889, 378899, 378999, 379999, 388888, 388889, 388899, 388999, 389999, 399999, 455555, 455557, 455558, 455559, 455577, 455578, 455579, 455588, 455589, 455599, 455777, 455778, 455779, 455788, 455789,
455799, 455888, 455889, 455899, 455999, 457777, 457778, 457779, 457788, 457789, 457799, 457888, 457889, 457899, 457999, 458888, 458889, 458899, 458999, 459999, 477777, 477778, 477779, 477788, 477789, 477799, 477888,
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 楼主| 发表于 2026-10-31 08:44 | 显示全部楼层
利用3#的思想——来道题。——求通项公式。谢谢各位网友！

a(n)=正整数(非0数码)分成2组,  2组积的差(最小)=n。

0=1-1。11=a(0)。
1=2-1。12=a(1)。
2=3-1。13=a(2)。
3=4-1。14=a(3)。
4=5-1。15=a(4)。
5=6-1。16=a(5)。
6=7-1。17=a(6)。
7=8-1。18=a(7)。
8=9-1。19=a(8)。
9=2*5-1。125=a(9)。
10=2*6-2。226=a(10)。
11=2*6-1。126=a(11)。
12=2*7-2。227=a(12)。
13=2*7-1。127=a(13)。
14=3*5-1。135=a(14)。
15=2*8-1。128=a(15)。
16=2*9-2。229=a(16)。
17=2*9-1。129=a(17)。
18=4*5-2。245=a(18)。
19=4*5-1。145=a(19)。
20=3*7-1。137=a(20)。
21=3*8-3。338=a(21)。
22=3*8-2。238=a(22)。
23=3*8-1。138=a(23)。
24=5*5-1。155=a(24)。
25=3*9-2。239=a(25)。
26=3*9-1。139=a(26)。
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