A239134 - OEIS

%I #23 Sep 17 2024 21:09:44

%S 1,6,7,6,2,6,3,4,5,1,1,1,1,1,1,1,1,1,1,6,1,4,2,3,2,4,2,4,3,7,1,2,3,3,

%T 2,2,3,5,2,6,1,8,4,4,2,2,2,2,2,2,1,2,2,2,2,4,2,4,3,6,1,3,5,6,2,4,3,2,

%U 3,3,1,3,2,6,2,3,2,6,2,4,1,2,4,4

%N Smallest k such that n^k contains k as a substring in its decimal representation.

%C It seems very likely a(n) < 10 for all n (even stronger, a(n) < 9 for all n).

%C It also seems very likely a(n) = {1,2,3} for sufficiently large n.

%C Counterexample: a(10^d - 2) = 6 for d >= 2. - _Robert Israel_, Sep 16 2024

%H Giovanni Resta, <a href="/A239134/b239134.txt">Table of n, a(n) for n = 1..10000</a>

%F a(A011531(k))=1, any k.

%F a(10*n) = a(n) if a(n) < 10. - _Robert Israel_, Sep 16 2024

%e 5^1 = 5 does not contain a 1 but 5^2 = 25 does contain a 2 so a(5) = 2.

%e 7^1 = 7 does not contain a 1, 7^2 = 49 does not contain a 2, but 7^3 = 343 does contain a 3 so a(7) = 3.

%p f:= proc(n) local k;

%p   for k from 1 to 9 do

%p     if member(k,convert(n^k,base,10)) then return k fi

%p   od;

%p   FAIL

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Sep 16 2024

%t a[n_] := Block[{k=1}, While[{} == StringPosition[ ToString[n^k], ToString[k]], k++]; k]; Array[a, 84] (* _Giovanni Resta_, Mar 11 2014 *)

%t sk[n_]:=Module[{k=1},While[SequenceCount[IntegerDigits[n^k],IntegerDigits[k]] == 0,k++];k]; Array[sk,90] (* _Harvey P. Dale_, May 12 2022 *)

%o (Python)

%o def Sub(x):

%o   for n in range(10**3):

%o     if str(x**n).find(str(n)) > -1:

%o       return n

%o x = 1

%o while x < 10**3:

%o   print(Sub(x))

%o   x += 1

%Y Cf. A045537, A061280.

%K nonn,base

%O 1,2

%A _Derek Orr_, Mar 10 2014