数字串的通项公式

王守恩

5^a(n) is smallest power of 5 beginning with n^2.——OEIS就没有了。

0, 11, 10, 69, 2, 38, 104, 87, 183, 93, 209, 232, 69, 298, 434, 477, 231, 377, 915, 483, 1413, 403, 74, 2077, 4, 1233, 1007, 1947, 1432, 2083, 1764, 2611, 2488, 1395, 1953, 1541, 644, 913, 5454, 6889, 946, 1411,
   
Table[Module[{t = IntegerDigits[n^2], k = 0}, While[Take[IntegerDigits[5^k], UpTo[Length[t]]] != t, k++]; k], {n, 49}]——众多个数字串可以共一个公式。

王守恩

对数运算不一定快。譬如——A059449——a(n)! begins with 2^n.——1, 2, 8, 14, 89, 164, 18, 266, 304, 292, 2522, 557, 26438, 27045, 57134, 21936。

AbsoluteTiming[Do[a = 2^n; b = Floor[N[Log[10, a]] + 1]; k = 1; While[c = Floor[N[Log[10, k!]] + 1]; a != (k! - Mod[k!, 10^(c - b)])/10^(c - b), k++]; Print[k], {n, 0, 15}]]——A059449——自带的公式。
{419.582, {1, 2, 8, 14, 89, 164, 18, 266, 304, 292, 2522, 557, 26438, 27045, 57134, 21936}}

AbsoluteTiming[Table[a = 2^n; b = IntegerLength[a]; k = 1; f = 1; While[True, f = f*k; c = IntegerLength[f]; If[c >= b, d = Quotient[f, 10^(c - b)]; If[d == a, Break[]]];  k++;]; k, {n, 0, 15}]]
{38.2066, {1, 2, 8, 14, 89, 164, 18, 266, 304, 292, 2522, 557, 26438, 27045, 57134, 21936}}

我这里出来就比A059449还要多了——{1, 2, 8, 14, 89, 164, 18, 266, 304, 292, 2522, 557, 26438, 27045, 57134, 21936, 224636, 529309}

王守恩

OEIS——A387487——Numbers in the ring of integers Z such that k * k = k, the idempotents of Z.——0,1——最短的数字串！！！——  "0"  与  "1"  。

KEYWORD——nonn,bref,fini,full,nice,core,new——AUTHOR——Peter Luschny—— Aug 31 2025。

王守恩

     有见过这数字串的,   给个通项公式。 谢谢！！！

{0, 1, 2, 3, 4, 55, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1155, 16, 17, 18, 19, 20, 21, 22, 23, 24, 2255, 26, 27, 28, 29, 30, 31, 32, 33, 34, 3355, 36, 37, 38, 39, 40, 41, 42, 43, 44, 4455, 46, 47, 48, 49, 50, 51, 52, 53, 54, 5555, 56, 57, 58, 59, 60,
61, 62, 63, 64, 6655, 66, 67, 68, 69, 70, 71, 72, 73, 74, 7755, 76, 77, 78, 79, 80, 81, 82, 83, 84, 8855, 86, 87, 88, 89, 90, 91, 92, 93, 94, 9955, 96, 97, 98, 99, 100, 101, 102, 103, 104, 110055, 106, 107, 108, 109, 110, 111, 112, 113, 114,
111155, 116, 117, 118, 119, 120, 121, 122, 123, 124, 112255, 126, 127, 128, 129, 130, 131, 132, 133, 134, 113355, 136, 137, 138, 139, 140, 141, 142, 143, 144, 114455, 146, 147, 148, 149, 150, 151, 152, 153, 154, 115555, 156, 157, 158,


补充内容 (2025-9-12 10:14):
个位为5时,所有数码加倍——Table[FromDigits[If[Mod[n, 10] == 5, {#, #}, #] & /@ IntegerDigits@n ;;;], {n, 0, 30}]

王守恩

接楼上。——OEIS不会有！！！

Table[FromDigits[Flatten[If[Mod[n, 10] == 5, {#, #}, #] & /@ IntegerDigits@n]], {n, 0, 30}]——个位为5时,所有数码加倍。

Table[FromDigits[Flatten[If[MemberQ[{3, 8}, Mod[n, 10]], {#, #}, #] & /@ IntegerDigits@n]],{n, 0, 30}]——个位为多个时，所有数码加倍。

Table[FromDigits[Flatten[If[# == 5, {#, #}, #] & /@ IntegerDigits@n]], {n, 0, 30}]——出现5时,就加倍。

Table[FromDigits[Flatten[If[MemberQ[{3, 8}, #], {#, #}, #] & /@ IntegerDigits@n]], {n, 0, 30}]——出现多个数码就加倍。

王守恩

A126933——给您配个通项公式——Table[SelectFirst[FromDigits /@ Tuples[{1, 2}, n], Mod[#, 2^n] == 0 &]/2^n, {n, 20}]

{1, 3, 14, 132, 691, 1908, 16579, 47352, 414301, 1183713, 5474669, 27151397, 135646011, 678174568, 6442602909, 18480090517, 85533990571, 424236721848, 4026815626549, 11550150977337}

王守恩

挺有意思的一串数——A023396——我来补个通项公式。

If any odd power of 2 ends with k 1's and 2's, they must be the first k terms of this sequence in reverse order.

2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1,
2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2,
2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2,

f[n_] := Module[{t, k}, If[n == 1, Return[2]]; t = f[n - 1]; k = t/2^(n - 1); If[OddQ[k], 1*10^(n - 1) + t, 2*10^(n - 1) + t]]
StringJoin[Riffle[ToString /@ Flatten[Reverse[IntegerDigits[f[276]]]], ", "]]

这个由递归算法生成的序列确实产生了一个无理数。它更可能是一个超越数。这个构造与钱珀瑙恩常数(Champernowne constant)类似, 但具有更复杂的自指涉结构。

王守恩

把 54(18×3) 个数字串摆在一起, 规律就来了。—— OEIS没有这样完整的。

A(2, 1) = {2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1}, {2, 12, 112, 2112, 22112, 122112, 2122112, 12122112, 212122112, 1212122112}, {1, 03, 014, 132, 0691, 01908, 16579, 047352, 0414301, 1183713},
A(2, 3) = {2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3}, {2, 32, 232, 3232, 23232, 223232, 2223232, 32223232, 232223232, 3232223232}, {1, 08, 029, 202, 0726, 03488, 17369, 125872, 0453561, 3156468},
A(2, 5) = {2, 5, 5, 5, 5, 2, 5, 5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 2, 5, 5, 5, 5, 2, 5, 5, 5, 2, 2, 5}, {2, 52, 552, 5552, 55552, 255552, 5255552, 55255552, 255255552, 2255255552}, {1, 13, 069, 347, 1736, 03993, 41059, 215842, 0498546, 2202398},
A(2, 7) = {2, 7, 2, 2, 2, 2, 7, 2, 7, 2, 7, 7, 7, 7, 2, 2, 7, 2, 7, 2, 7, 7, 2, 7, 7, 2, 7, 2, 7, 7}, {2, 72, 272, 2272, 22272, 222272, 7222272, 27222272, 727222272, 2727222272}, {1, 18, 034, 142, 0696, 03473, 56424, 106337, 1420356, 2663303},
A(2, 9) = {2, 9, 9, 2, 9, 2, 2, 2, 2, 2, 2, 9, 9, 2, 9, 9, 9, 9, 2, 2, 9, 2, 9, 2, 9, 2, 2, 9, 2, 2}, {2, 92, 992, 2992, 92992, 292992, 2292992, 22292992, 222292992, 2222292992}, {1, 23, 124, 187, 2906, 04578, 17914, 087082, 0434166, 2170208},
A(4, 1) = {4, 4, 1, 4, 1, 4, 1, 4, 4, 1, 1, 4, 4, 4, 4, 1, 4, 4, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 4, 4}, {4, 44, 144, 4144, 14144, 414144, 1414144, 41414144, 441414144, 1441414144}, {2, 11, 018, 259, 0442, 06471, 11048, 161774, 0862137, 1407631},
A(4, 3) = {4, 4, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 4, 3, 4, 4, 3, 3, 4, 3}, {4, 44, 344, 3344, 33344, 433344, 3433344, 33433344, 333433344, 3333433344}, {2, 11, 043, 209, 1042, 06771, 26823, 130599, 0651237, 3255306},
A(4, 5) = {4, 4, 5, 4, 4, 4, 4, 5, 4, 5, 4, 5, 5, 5, 5, 4, 4, 5, 4, 5, 4, 5, 5, 4, 5, 5, 4, 5, 4, 5}, {4, 44, 544, 4544, 44544, 444544, 4444544, 54444544, 454444544, 5454444544}, {2, 11, 068, 284, 1392, 06946, 34723, 212674, 0887587, 5326606},
A(4, 7) = {4, 4, 7, 7, 4, 4, 4, 4, 4, 4, 7, 4, 4, 4, 4, 7, 7, 4, 7, 7, 7, 4, 7, 4, 4, 7, 4, 7, 4, 4}, {4, 44, 744, 7744, 47744, 447744, 4447744, 44447744, 444447744, 4444447744}, {2, 11, 093, 484, 1492, 06996, 34748, 173624, 0868062, 4340281},
A(4, 9) = {4, 4, 9, 4, 9, 9, 4, 9, 4, 9, 4, 4, 9, 9, 4, 9, 4, 9, 9, 9, 4, 4, 4, 4, 9, 4, 4, 4, 4, 9}, {4, 44, 944, 4944, 94944, 994944, 4994944, 94994944, 494994944, 9494994944}, {2, 11, 118, 309, 2967, 15546, 39023, 371074, 0966787, 9272456},
A(6, 1) = {6, 1, 6, 1, 1, 1, 6, 1, 6, 1, 6, 6, 1, 6, 6, 6, 6, 6, 6, 1, 6, 1, 1, 1, 6, 6, 6, 6, 6, 6}, {6, 16, 616, 1616, 11616, 111616, 6111616, 16111616, 616111616, 1616111616}, {3, 04, 077, 101, 0363, 01744, 47747, 062936, 1203343, 1578234},
A(6, 5) = {6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 5, 6, 5, 5, 6, 5, 5, 5, 5, 6, 5, 5, 6, 6, 5, 6}, {6, 56, 656, 6656, 66656, 566656, 6566656, 66566656, 666566656, 6666566656}, {3, 14, 082, 416, 2083, 08854, 51302, 260026, 1301888, 6510319},
A(6, 7) = {6, 7, 7, 7, 6, 6, 6, 6, 7, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, 6, 7}, {6, 76, 776, 7776, 67776, 667776, 6667776, 66667776, 766667776, 6766667776}, {3, 19, 097, 486, 2118, 10434, 52092, 260421, 1497398, 6608074},
A(8, 1) = {8, 8, 8, 1, 8, 1, 8, 1, 1, 8, 8, 8, 8, 8, 8, 1, 8, 1, 8, 8, 8, 8, 1, 8, 1, 8, 1, 1, 8, 1}, {8, 88, 888, 1888, 81888, 181888, 8181888, 18181888, 118181888, 8118181888}, {4, 22, 111, 118, 2559, 02842, 63921, 071023, 0230824, 7927912},
A(8, 3) = {8, 8, 8, 3, 3, 3, 3, 8, 3, 3, 3, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 3, 8, 8, 3, 3, 8, 8}, {8, 88, 888, 3888, 33888, 333888, 3333888, 83333888, 383333888, 3383333888}, {4, 22, 111, 243, 1059, 05217, 26046, 325523, 0748699, 3304037},
A(8, 5) = {8, 8, 8, 5, 8, 8, 8, 5, 5, 8, 5, 8, 5, 8, 5, 5, 5, 8, 8, 8, 8, 5, 5, 5, 8, 5, 8, 5, 5, 5}, {8, 88, 888, 5888, 85888, 885888, 8885888, 58885888, 558885888, 8558885888}, {4, 22, 111, 368, 2684, 13842, 69421, 230023, 1091574, 8358287},
A(8, 7) = {8, 8, 8, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 7, 8, 8, 8, 7, 8, 8, 8, 8, 8, 7, 7, 8, 8, 7, 8}, {8, 88, 888, 7888, 77888, 877888, 7877888, 87877888, 787877888, 8787877888}, {4, 22, 111, 493, 2434, 13717, 61546, 343273, 1538824, 8581912},
A(8, 9) = {8, 8, 8, 9, 8, 9, 9, 8, 9, 8, 9, 8, 8, 9, 9, 8, 9, 8, 9, 9, 9, 8, 8, 8, 8, 9, 8, 8, 8, 8}, {8, 88, 888, 9888, 89888, 989888, 9989888, 89989888, 989989888, 8989989888}, {4, 22, 111, 618, 2809, 15467, 78046, 351523, 1933574, 8779287},

f[n_] := Module[{t, k}, If[n == 1, Return[8]]; t = f[n - 1]; k = t/2^(n - 1); If[OddQ[k], 5*10^(n - 1) + t, 8*10^(n - 1) + t]]
StringJoin[Riffle[ToString /@ Flatten[Reverse[IntegerDigits[f[30]]]], ", "]]——第 1 列数字串的通项公式。

Table[SelectFirst[FromDigits /@ Tuples[{5, 8}, n], Mod[#, 2^n] == 0 &], {n, 10}]——第 2 列数字串的通项公式。

Table[SelectFirst[FromDigits /@ Tuples[{5, 8}, n], Mod[#, 2^n] == 0 &]/2^n, {n, 10}]——第 3 列数字串的通项公式。

第 1 列数字串是个“无理数”——差不离。第 1 列数字串是个“超越数”——没有认定。

王守恩

顶点是三角形数的三角形图——图见A062708——记"0"关于"n"的对称数为"a(n)"。譬如
a(1)=11,
a(2)=13,
a(3)=15,
a(4)=17,
a(5)=19,
a(6)=21,
a(7)=23,
a(8)=25,
a(9)=27,
a(10)=56,
a(11)=58,
a(12)=60,
a(13)=62,
a(14)=64,
a(15)=66,
a(16)=68,
......
得到一串数——11, 13, 15, 17, 19, 21, 23, 25, 27, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 137, 139, 141, ......

a(n) = 2 n + 9 Round[Sqrt[2 (n + 1)]/3]^2。——OEIS没有这串数。

王守恩

20个数字串。——a(n) contains n digits (either '2' or '3') and is divisible by 2^n.

A(2, 1) = {2, 12, 112, 2112, 22112, 122112, 2122112, 12122112, 212122112, 1212122112}, ——A053312——2025。
A(2, 3) = {2, 32, 232, 3232, 23232, 223232, 2223232, 32223232, 232223232, 3232223232}, ——A053316——2019。
A(2, 5) = {2, 52, 552, 5552, 55552, 255552, 5255552, 55255552, 255255552, 2255255552}, ——A053317——2022。
A(2, 7) = {2, 72, 272, 2272, 22272, 222272, 7222272, 27222272, 727222272, 2727222272}, ——A053318——2025。
A(2, 9) = {2, 92, 992, 2992, 92992, 292992, 2292992, 22292992, 222292992, 2222292992}, ——A053313——2015。
A(4, 1) = {4, 44, 144, 4144, 14144, 414144, 1414144, 41414144, 441414144, 1441414144}, ——A053314——2022。
A(4, 3) = {4, 44, 344, 3344, 33344, 433344, 3433344, 33433344, 333433344, 3333433344}, ——A035014——2017。
A(4, 5) = {4, 44, 544, 4544, 44544, 444544, 4444544, 54444544, 454444544, 5454444544}, ——A053315——2019。
A(4, 7) = {4, 44, 744, 7744, 47744, 447744, 4447744, 44447744, 444447744, 4444447744}, ——A053332——2017。
A(4, 9) = {4, 44, 944, 4944, 94944, 994944, 4994944, 94994944, 494994944, 9494994944}, ——A053333——2023。
A(6, 1) = {6, 16, 616, 1616, 11616, 111616, 6111616, 16111616, 616111616, 1616111616}, ——A053334——2025。
A(6, 3) = {6, 36, 336, 6336, 66336, 366336, 6366336, 36366336, 636366336, 3636366336}, ——A053335——2000。
A(6, 5) = {6, 56, 656, 6656, 66656, 566656, 6566656, 66566656, 666566656, 6666566656}, ——A053336——2020。
A(6, 7) = {6, 76, 776, 7776, 67776, 667776, 6667776, 66667776, 766667776, 6766667776}, ——A053337——2016。
A(6, 9) = {6, 96, 696, 9696, 69696, 669696, 6669696, 96669696, 696669696, 9696669696}, ——A053338——2023。
A(8, 1) = {8, 88, 888, 1888, 81888, 181888, 8181888, 18181888, 118181888, 8118181888}, ——A053376——2017。
A(8, 3) = {8, 88, 888, 3888, 33888, 333888, 3333888, 83333888, 383333888, 3383333888}, ——A053377——2015。
A(8, 5) = {8, 88, 888, 5888, 85888, 885888, 8885888, 58885888, 558885888, 8558885888}, ——A053378——2000。
A(8, 7) = {8, 88, 888, 7888, 77888, 877888, 7877888, 87877888, 787877888, 8787877888}, ——A053379——2000。
A(8, 9) = {8, 88, 888, 9888, 89888, 989888, 9989888, 89989888, 989989888, 8989989888}, ——A053380——2000。