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

Author

Erich Friedman

Keywords

• base
• nonn

Links

• 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

Crossrefs

Cf. A051248, A070327, A104782.

Programs

Haskell

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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