A104782 -id:A104782 - 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).

A051248 n^a(n) is the smallest power of n (with a(n) > 1) which starts with n.

Original entry on oeis.org

2, 8, 18, 6, 24, 10, 20, 11, 22, 2, 26, 14, 10, 8, 92, 6, 166, 5, 19, 11, 60, 39, 48, 51, 94, 42, 467, 86, 14, 66, 58, 3, 5268, 33, 58, 10, 45, 70, 23, 6, 205, 70, 31, 15, 50, 84, 248, 70, 72, 94, 107, 170, 30, 10139, 182, 136, 87, 318, 49, 10, 178, 237, 295, 99, 17, 206

Offset: 1

Michel ten Voorde

A070327(n) = n^a(n) for n > 1. - Reinhard Zumkeller, Sep 29 2011

The largest of the first 10000 terms is a(1955) = 119589011. - Giovanni Resta, May 27 2016

			2^8 = 256 begins with 2, and 2^k does not begin with 2 for any smaller integer k > 1, so a(2) = 8.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 275 terms from Reinhard Zumkeller)

Cf. A045537, A104782.

import Data.List (isPrefixOf)
a051248 n = 2 + length
   (takeWhile (not . (show n `isPrefixOf`) . show) $ iterate (* n) (n^2))
-- Reinhard Zumkeller, Sep 29 2011

f[n_]:=Module[{i=2},While[Take[IntegerDigits[n^i],IntegerLength[n]] != IntegerDigits[n],i++];i]; Array[f,70] (* Harvey P. Dale, Oct 04 2011 *)

from itertools import count
def a(n):
    s = str(n)
    return next(e for e in count(2) if str(n**e).startswith(s))
print([a(n) for n in range(1, 33)]) # Michael S. Branicky, Feb 23 2025

More terms from Jud McCranie

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.

A373337 Records in A045537.

Original entry on oeis.org

2, 5, 11, 14, 26, 28, 31, 50, 58, 59, 71, 72, 98, 107, 148, 166, 170, 172, 173, 211, 221, 223, 546, 549, 601, 616, 704, 716, 884, 1152, 1774, 1826, 1847, 1976, 1980, 2213, 2494, 3561, 4587, 4615, 4691, 5110, 5196, 5790, 5810, 6070, 6198, 6255, 6648, 6655, 6697

Offset: 1

Gonzalo Martínez, Jun 01 2024

A045537(n) is the exponent of the least perfect power of n containing n as a substring. As n grows, larger integers are recorded.

			a(1) = A045537(0) = 2.
a(2) = A045537(2) = 5.
a(3) = A045537(11) = 11.
a(4) = A045537(12) = 14.
a(5) = A045537(15) = 26.
a(6) = A045537(102) = 28.

Cf. A045537, A104782, A111442, A373942.

a(33)-a(37) from Michel Marcus, Jun 04 2024
a(38)-a(49) from Michael S. Branicky, Jun 05 2024
a(50)-a(51) from Jinyuan Wang, Jun 17 2025

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

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A051248 n^a(n) is the smallest power of n (with a(n) > 1) which starts with n.

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

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Python

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A373337 Records in A045537.

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