数字串的通项公式

王守恩 · 发表于 2025-8-12 17:56:01

这些数字串可都是OEIS没有的。

a(n)= n^2——{1, 4, 9, 13, 17, 23, 29, 37, 45, 55, 65, 77, 89, 102, 111, 122, 133, 144, 157, 170, 183, 198, 213, 228, 245, 262, 279, 298, 317, 336, 357, 378, 399, 422, 445, 468, 493, 518, 543, 570, 597, 624, 653, 682,——
a(n)=10^n——{1, 10, 55, 370, 2777, 22222, 185185, 1587301, 13888888, 123456790, 1111111111, 10101010100, 91919191919, 842592592592, 7777777777777, 72222222222222, 674074074074074, 6319444444444444,——
a(n)= n^n——{1, 4, 18, 122, 1058, 11553, 155775, 2555475, 49816449, 1111111111, 26947525611, 752267629947, 22427586978811, 748207862444608, 26412058911292381, 1029360799201087511, 41917568649872393764,——
a(n)=n(n+1)/2——{1, 3, 6, 10, 12, 15, 19, 23, 27, 32, 38, 44, 50, 57, 65, 73, 81, 90, 100, 106, 113, 121, 128, 136, 145, 153, 162, 172, 181, 191, 202, 212, 223, 235, 246, 258, 271, 283, 296, 310, 323, 337, 352, 366, 381, 397, 412, ——
Table[Module[{t = k = 0}, While[t < n (n + 1)/2, k++; t += IntegerLength[k];]; k], {n, 50}]
{1},
{1, 2, 3},
{1, 2, 3, 4, 5, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, ...,4, 8, 4, 9, 5, 0},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, ...,2, 5, 3, 5, 4, 5, 5, 5, 6, 5, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, ...,7, 5, 8, 5, 9, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, ...,6, 3, 6, 4, 6, 5, 6, 6, 6, 7, 6, 8, 6, 9, 7, 0, 7, 1, 7, 2, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, ...,6, 9, 7, 0, 7, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 7, 7, 8, 7, 9, 8, 0, 8, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, ...,5, 7, 6, 7, 7, 7, 8, 7, 9, 8, 0, 8, 1, 8, 2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 8, 9, 9, 0},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, ...,2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 8, 9, 9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, ...,9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 4, 1, 0, 5, 1, 0, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, ...,9, 8, 9, 9, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 4, 1, 0, 5, 1, 0, 6, 1, 0, 7, 1, 0, 8, 1, 0, 9, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3},
Table[First[RealDigits[ChampernowneNumber[], 10, n (n + 1)/2]], {n, 21}]

王守恩 · 发表于前天 11:29

northwolves 发表于 2025-7-27 08:14

题目1。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 24579, 235689, 2345789, 13456789, 123456789, 1234567899, 12344567899, 123345667899, 1233455677899, 12234456677899, 122334556678899, 1223344556778899, 12233445566778899, 112233445566778899}
Table[s = Sort[PadRight[{}, n^2, Range[9]]]; u = Table[{s[]}, {i, n}]; r = s[[n + 1 ;;]]; For[i = 1, i <= Length[r], i++, t = r[]; w = First[Ordering[FromDigits /@ u, 1]]; u[[w]] = Append[u[[w]], t];]; Max[FromDigits /@ u], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目2。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 7642, 77542, 886532, 8865432, 88765431, 987654321, 9876654321, 98776543321, 988765543321, 9887665443221, 98877655433221, 988876655433221, 9888776554433211, 98887766554433211, 998877665544332211}
Table[s = Reverse[Sort[PadRight[{}, n^2, Range[9]]]]; u = Table[{s[]}, {i, n}]; r = s[[n + 1 ;;]]; For[i = 1, i <= Length[r], i++, t = r[]; w = First[Ordering[FromDigits /@ u, 1]]; u[[w]] = Append[u[[w]], t];]; Min[FromDigits /@ u], {n, 1, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目3。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 23579, 234679, 2245789, 22345789, 123456789, 1023456789, 10234567899, 102344567899, 1023345667899, 10223445677899, 101233455677899, 1012334456678899, 10122344556778899, 101223344566778899}
Table[s = Sort[Mod[Range[n^2], 10]]; t = Count[s, 0]; v = s[[t + 1 ;; t + n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, Join[s[[1 ;; t]], s[[t + n + 1 ;;]]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目4。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 6642, 77531, 875421, 8865421, 88754321, 887654321, 9876543210, 98765543210, 987765432210, 9877665433210, 98876554432210, 988766554332110, 9887766544322110, 98887665544322110, 988877655443322110}
Table[t = Sort[Mod[Range[n^2], 10], Greater][[1 ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, Sort[Mod[Range[n^2], 10], Greater][[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

这4个编码是不是搞复杂了？谢谢！！！

王守恩 · 发表于前天 14:54

题目1。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 24579, 235689, 2345789, 13456789, 123456789, 1234567899, 12344567899, 123345667899, 1233455677899, 12234456677899, 122334556678899, 1223344556778899, 12233445566778899, 112233445566778899}
Table[s = Sort[Mod[Range[0, n^2 - 1], 9] + 1]; t = Count[s, 0]; v = s[[t + 1 ;; t + n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, Join[s[[1 ;; t]], s[[t + n + 1 ;;]]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目2。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 7642, 77542, 886532, 8865432, 88765431, 987654321, 9876654321, 98776543321, 988765543321, 9887665443221, 98877655433221, 988876655433221, 9888776554433211, 98887766554433211, 998877665544332211}
Table[s = Sort[Mod[Range[0, n^2 - 1], 9] + 1, Greater]; t = s[[1 ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目3。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 23579, 234679, 2245789, 22345789, 123456789, 1023456789, 10234567899, 102344567899, 1023345667899, 10223445677899, 101233455677899, 1012334456678899, 10122344556778899, 101223344566778899}
Table[s = Sort[Mod[Range[n^2], 10]]; t = Count[s, 0]; v = s[[t + 1 ;; t + n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, Join[s[[1 ;; t]], s[[t + n + 1 ;;]]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目4。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 6642, 77531, 875421, 8865421, 88754321, 887654321, 9876543210, 98765543210, 987765432210, 9877665433210, 98876554432210, 988766554332110, 9887766544322110, 98887665544322110, 988877655443322110}
Table[s = Sort[Mod[Range[n^2], 10], Greater]; t = s[[1 ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

这4个编码是不是搞复杂了？——这些按钮我还是学不好。谢谢！！！

王守恩 · 发表于前天 16:25

题目1。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 24579, 235689, 2345789, 13456789, 123456789, 1234567899, 12344567899, 123345667899, 1233455677899, 12234456677899, 122334556678899, 1223344556778899, 12233445566778899, 112233445566778899}
Table[s = Sort[Mod[Range[0, n^2 - 1], 9] + 1]; v = s[[ ;; n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目2。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 7642, 77542, 886532, 8865432, 88765431, 987654321, 9876654321, 98776543321, 988765543321, 9887665443221, 98877655433221, 988876655433221, 9888776554433211, 98887766554433211, 998877665544332211}
Table[s = Sort[Mod[Range[0, n^2 - 1], 9] + 1, Greater]; t = s[[ ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目3。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 23579, 234679, 2245789, 22345789, 123456789, 1023456789, 10234567899, 102344567899, 1023345667899, 10223445677899, 101233455677899, 1012334456678899, 10122344556778899, 101223344566778899}
Table[s = Sort[Mod[Range[n^2], 10]]; t = Count[s, 0]; v = s[[t + 1 ;; t + n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, Join[s[[1 ;; t]], s[[t + n + 1 ;;]]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目4。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 6642, 77531, 875421, 8865421, 88754321, 887654321, 9876543210, 98765543210, 987765432210, 9877665433210, 98876554432210, 988766554332110, 9887766544322110, 98887665544322110, 988877655443322110}
Table[s = Sort[Mod[Range[n^2], 10], Greater]; t = s[[ ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

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[原创] 数字串的通项公式