A051248 -id:A051248 - cp's OEIS frontend

A045537 Least nontrivial exponent e such that n is a substring of n^e.

2, 2, 5, 5, 3, 2, 2, 5, 5, 3, 2, 11, 14, 10, 8, 26, 6, 17, 5, 11, 5, 6, 10, 15, 3, 2, 19, 15, 7, 8, 5, 11, 3, 14, 14, 10, 6, 10, 6, 11, 3, 6, 18, 5, 11, 5, 18, 9, 5, 3, 2, 3, 7, 16, 17, 11, 3, 5, 9, 11, 2, 6, 7, 7, 11, 17, 15, 8, 5, 11, 5, 9, 8, 5, 8, 3, 2, 16, 21, 11, 5, 6, 14, 4, 11, 22, 22, 7

Offset: 0

Erich Friedman

Reinhard Zumkeller and Zak Seidov, Table of n, a(n) for n = 0..10000. Terms up to a(1000) from Reinhard Zumkeller.
Code Golf StackExchange, Self-contained powers, coding challenge started Nov 28 2018.
Index entries for sequences related to automorphic numbers

Cf. A051248, A070327, A104782.

import Data.List (isInfixOf)
a045537 n = 2 + length
   (takeWhile (not . ((show n) `isInfixOf`) . show) $ iterate (* n) (n^2))
-- Reinhard Zumkeller, Sep 29 2011

f[n_] := Block[{k = 2}, While[ StringPosition[ ToString[n^k], ToString[n]] == {}, k++ ]; k]; Table[ f[n], {n, 0, 87}] (* Robert G. Wilson v, May 09 2005 *)

a(n) = my(s = Str(n), k=2); while (#strsplit(Str(n^k), s) == 1, k++); k; \\ Michel Marcus, Jun 04 2024

from itertools import count
def a(n):
    s = str(n)
    return next(e for e in count(2) if s in str(n**e))
print([a(n) for n in range(88)]) # Michael S. Branicky, Feb 23 2025

n^a(n) = A104782(n).

A104782 Smallest n^e (e>1) containing n in decimal representation.

Original entry on oeis.org

1, 32, 243, 64, 25, 36, 16807, 32768, 729, 100, 285311670611, 1283918464548864, 137858491849, 1475789056, 3787675244106352329254150390625, 16777216, 827240261886336764177, 1889568, 116490258898219, 3200000, 85766121

Offset: 1

Reinhard Zumkeller, Mar 25 2005

a(n) = n^A045537(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A051248, A070327.

from itertools import count
def a(n):
    s = str(n)
    return next(n**e for e in count(2) if s in str(n**e))
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 23 2025

A070327 Smallest nontrivial power of n starting with n and greater than n.

Original entry on oeis.org

256, 387420489, 4096, 59604644775390625, 60466176, 79792266297612001, 8589934592, 984770902183611232881, 100, 1191817653772720942460132761, 1283918464548864, 137858491849, 1475789056

Offset: 2

Amarnath Murthy, May 11 2002

a(n) = n ^ A051248(n). - Reinhard Zumkeller, Sep 29 2011

Reinhard Zumkeller, Table of n, a(n) for n = 2..32

Cf. A045537, A104782.

from itertools import count
def a(n):
    s = str(n)
    return next(n**e for e in count(2) if str(n**e).startswith(s))
print([a(n) for n in range(2, 16)]) # Michael S. Branicky, Feb 23 2025

f[n_] := Block[{k = 2, s = ToString[n]}, While[ StringPosition[ ToString[n^k], s, 1] != {{1, Integer_}}, k++ ]; n^k]; Table[ f[n], {n, 2, 10}]

Edited by Robert G. Wilson v, May 14 2002

The 15th term is too large to include.

A073601 Least k>1 such that n^k and n have equal leading decimal digits.

Original entry on oeis.org

2, 8, 18, 6, 24, 10, 20, 11, 22, 2, 2, 2, 2, 2, 6, 5, 5, 4, 4, 8, 20, 4, 4, 9, 6, 8, 8, 3, 3, 18, 48, 3, 3, 3, 12, 10, 8, 6, 6, 6, 6, 9, 20, 15, 4, 4, 4, 20, 14, 24, 18, 8, 19, 16, 5, 5, 34, 18, 10, 10, 15, 25, 6, 6, 17, 12, 7, 7, 26, 20, 21, 8, 23, 24, 9, 18, 10, 29

Offset: 1

Reinhard Zumkeller, Aug 04 2002

A073600(n) = n^a(n);

A000030(n^a(n)) = A000030(n).

			a(4)=6, as 4^6=4096=A073600(4) is the least power of 4 with initial digit =4: 4^2=16, 4^3=64, 4^4=256 and 4^5=1024.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A051248.

a073601 n = 2 + length
   (takeWhile ((a000030 n /=) . a000030) $ iterate (* n) (n^2))
-- Reinhard Zumkeller, Sep 27 2011

fi[n_] := First[IntegerDigits[n]]; Table[k = 2; While[fi[n^k] != fi[n], k++]; k, {n, 78}] (* Jayanta Basu, Jul 02 2013 *)

A068001 Smallest number whose n-th power (A067457) starts with n.

Original entry on oeis.org

1, 5, 7, 8, 9, 2, 5, 13, 46, 2, 19, 15, 6, 74, 12, 58, 4, 8, 5, 52, 6, 128, 85, 53, 15, 61, 113, 17, 55, 52, 83, 47, 147, 43, 297, 238, 63, 117, 51, 49, 145, 98, 219, 109, 48, 38, 114, 283, 423, 226, 334, 168, 38, 87, 91, 58, 189, 166, 42, 222, 92, 59, 133, 86, 544, 264, 5

Offset: 1

Robert G. Wilson v, Feb 08 2002

			a(3) = 7 because 7^3 = 343 which starts with 3 and there is no number less than 7 with this property.

Paolo P. Lava, Table of n, a(n) for n = 1..500

Cf. A051248, A067457.

Do[k = Floor[ Log[ 10, n] + 1]; While[ FromDigits[ Take[ IntegerDigits [k^n], Floor[ Log[ 10, n] + 1]]] != n, k++ ]; Print[k^n], {n, 1, 100} ]

A177047 Least k>1 such that n^k starts with n in base 2.

Original entry on oeis.org

2, 2, 3, 2, 8, 3, 6, 2, 7, 20, 12, 8, 4, 6, 11, 2, 13, 7, 86, 26, 19, 12, 3, 13, 15, 11, 5, 6, 8, 22, 22, 2, 24, 24, 9, 7, 25, 106, 348, 29, 15, 24, 55, 25, 60, 87, 93, 13, 84, 15, 4, 11, 12, 5, 33, 58, 324, 8, 18, 33, 44, 66, 45, 2, 46, 24, 92, 24, 38, 32, 128, 7, 59, 44, 36, 114, 16, 404

Offset: 1

Vladimir Reshetnikov, May 01 2010

a(639) > 10^5. - Seiichi Manyama, Dec 22 2019

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A051248, A177048.

toBinary 0 = []
toBinary n = toBinary (n `div` 2) ++ [odd n]
a = [2 + fromJust (findIndex (isPrefixOf (toBinary n)) [toBinary (n ^ k) | k <- [2..]]) | n <- [1..]]

a[n_] := For[k = 2, True, k++, If[ MatchQ[ IntegerDigits[n^k, 2], {Sequence @@ IntegerDigits[n, 2], _}], Return[k]]]; a /@ Range[59] (* Jean-François Alcover, Jun 05 2013 *)

a(n)=my(b=binary(n),k=1);while(binary(n^k++)[1..#b]!=b,);k \\ Charles R Greathouse IV, Jun 06 2013

More terms from Seiichi Manyama, Dec 22 2019

A273007 a(n) is the smallest exponent > 1 such that p^a(n) ends with p, where p is the n-th prime.

Original entry on oeis.org

5, 5, 2, 5, 11, 21, 21, 11, 21, 11, 11, 21, 6, 5, 21, 21, 11, 6, 21, 11, 21, 11, 21, 11, 21, 11, 101, 21, 51, 101, 101, 51, 101, 51, 11, 11, 21, 101, 101, 101, 51, 51, 51, 5, 101, 11, 51, 101, 101, 51, 101, 51, 26, 3, 21, 101, 51, 51, 101, 26, 101, 21, 5, 51

Offset: 1

Paolo P. Lava, May 24 2016

			2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32;
3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A051248, A115739, A273527.

P:=proc(q) local d,k,n; for n from 1 to q do if isprime(n) then d:=ilog10(n)+1;
for k from 2 to q do if n=(n^k mod 10^d) then print(k); break; fi; od; fi; od; end: P(10^3);

Table[Length[NestWhileList[p #&,p^2,Mod[#,10^IntegerLength[p]]!=p&]]+1,{p,Prime[ Range[65]]}] (* Harvey P. Dale, Jul 25 2019 *)

A273527 a(n) is the smallest exponent > 1 such that p^a(n) begins with p, where p is the n-th prime.

Original entry on oeis.org

8, 18, 24, 20, 26, 10, 166, 19, 48, 14, 58, 45, 205, 31, 248, 30, 49, 178, 3054, 122, 140, 294, 174, 80, 152, 233, 79, 920, 295, 359, 107, 308, 257, 8, 180, 96, 98, 34, 230, 921, 527, 164, 428, 901, 344, 88, 627, 1003, 192, 240, 50, 38, 1747, 609, 1028, 432, 122

Offset: 1

Paolo P. Lava, May 24 2016

Subset of A051248.

			2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64, 2^7 = 128, 2^8 = 256;
3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243, ..., 3^18 = 387420489.

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A051248, A273007.

P:=proc(q) local b,d,k,n; for n from 1 to q do if isprime(n) then d:=ilog10(n);
for k from 2 to q do b:=ilog10(n^k); if n=trunc(n^k/10^(b-d)) then print(k);
break; fi; od;  fi; od; end: P(10^6);

sep[n_]:=Module[{len=IntegerLength[n],idn=IntegerDigits[n],exp=2}, While[ Take[ IntegerDigits[ n^exp],len] != idn,exp++];exp]; sep/@Prime[ Range[ 60]] (* Harvey P. Dale, Jun 15 2016 *)

a(n) = A051248(A000040(n)).

A378857 a(n) is the smallest k > 1 such that n^k starts and ends with n, or -1 if there is no such k.

Original entry on oeis.org

2, 21, 41, 11, 24, 10, 33, 73, 153, -1, 171, 241, 361, -1, -1, 6, 461, -1, 471, -1, 12086, -1, 1281, 51, 94, -1, 1181, 701, 1091, -1, 231, 197, 5781, -1, -1, 161, 221, -1, 1231, -1, 236, -1, 61, 1451, -1, -1, 861, 13381, 143, -1, 107, 501, 2761, -1, -1, 136

Offset: 1

Gonzalo Martínez, Feb 10 2025

Given an integer n such that a(n) != -1, it follows that n^A051248(n) starts with n and n^A074250(n) ends with n, and both are the smallest exponents that satisfy these conditions. In this case, n^a(n) starts and ends with n. Due to this condition, larger exponents are required. For example, for n = 21 the smallest exponent satisfying the condition is 12086.

			2^21 = 2097152 is the smallest power of 2 that starts and ends with 2, so a(2) = 21.

Cf. A051248, A074250, A374026.

a(n) >= max{A074250(n), A051248(n)} except that a(n) = -1 when A074250(n) = -1.

A343594 Numbers k that, when written in all bases from base 2 to base 10, are a substring of k^k when written in the same base.

Original entry on oeis.org

1, 5, 17, 25, 31, 41, 63, 92, 151, 170, 202, 221, 263, 266, 278, 322, 327, 347, 364, 401, 404, 412, 421, 423, 437, 467, 470, 482, 490, 498, 501, 515, 519, 543, 558, 578, 590, 612, 623, 636, 646, 647, 671, 683, 685, 705, 707, 717, 718, 726, 764, 785, 795, 859, 867, 872, 875, 881, 890, 892, 897

Offset: 1

Scott R. Shannon, Apr 21 2021

			5 is a term. See below table:
.
   base  |  5 in base  |  5^5 in base
---------+-------------+-------------
    10          5                3125
     9          5                4252
     8          5                6065
     7          5               12053
     6          5               22245
     5         10              100000
     4         11              300311
     3         12            11021202
     2        101        110000110101
.
5^5 in all bases contains 5 in that base as a substring.

Cf. A049329, A045537, A051248.

str(v) = my(s=""); for (k=1, #v, s = concat(s, Str(v[k]))); s;
isok(k) = {for (b=2, 10, my(kb = digits(k, b), kkb = digits(k^k, b)); if (#strsplit(str(kkb), str(kb)) <=1 , return (0));); return (1);} \\ Michel Marcus, Apr 26 2021

from sympy.ntheory import digits
def nstr(n, b): return "".join(map(str, digits(n, b=b)[1:]))
def ok(k): return all(nstr(k, b) in nstr(k**k, b) for b in range(10, 1, -1))
print(list(filter(ok, range(900)))) # Michael S. Branicky, Apr 25 2021

cp's OEIS Frontend

A045537 Least nontrivial exponent e such that n is a substring of n^e.

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A104782 Smallest n^e (e>1) containing n in decimal representation.

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A070327 Smallest nontrivial power of n starting with n and greater than n.

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A073601 Least k>1 such that n^k and n have equal leading decimal digits.

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Haskell

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A068001 Smallest number whose n-th power (A067457) starts with n.

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Mathematica

A177047 Least k>1 such that n^k starts with n in base 2.

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A273007 a(n) is the smallest exponent > 1 such that p^a(n) ends with p, where p is the n-th prime.

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A273527 a(n) is the smallest exponent > 1 such that p^a(n) begins with p, where p is the n-th prime.

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