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

%I A045537 #43 Feb 23 2025 09:25:31
%S A045537 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,
%T A045537 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,
%U A045537 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
%N A045537 Least nontrivial exponent e such that n is a substring of n^e.
%H A045537 Reinhard Zumkeller and Zak Seidov, <a href="/A045537/b045537.txt">Table of n, a(n) for n = 0..10000</a>. Terms up to a(1000) from Reinhard Zumkeller.
%H A045537 Code Golf StackExchange, <a href="https://codegolf.stackexchange.com/questions/176734/self-contained-powers">Self-contained powers</a>, coding challenge started Nov 28 2018.
%H A045537 <a href="/index/Ar#automorphic">Index entries for sequences related to automorphic numbers</a>
%F A045537 n^a(n) = A104782(n).
%t A045537 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 *)
%o A045537 (Haskell)
%o A045537 import Data.List (isInfixOf)
%o A045537 a045537 n = 2 + length
%o A045537    (takeWhile (not . ((show n) `isInfixOf`) . show) $ iterate (* n) (n^2))
%o A045537 -- _Reinhard Zumkeller_, Sep 29 2011
%o A045537 (PARI) a(n) = my(s = Str(n), k=2); while (#strsplit(Str(n^k), s) == 1, k++); k; \\ _Michel Marcus_, Jun 04 2024
%o A045537 (Python)
%o A045537 from itertools import count
%o A045537 def a(n):
%o A045537     s = str(n)
%o A045537     return next(e for e in count(2) if s in str(n**e))
%o A045537 print([a(n) for n in range(88)]) # _Michael S. Branicky_, Feb 23 2025
%Y A045537 Cf. A051248, A070327, A104782.
%K A045537 base,nonn
%O A045537 0,1
%A A045537 _Erich Friedman_