A253600 Smallest exponent k>1 such that n and n^k have some digits in common.

2, 2, 5, 5, 3, 2, 2, 5, 5, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 2, 5, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Author

Michel Marcus, Jan 05 2015

Keywords

• nonn
• base

Comments

For all n, n^5-n is divisible by 10, and so n^5 == n (mod 10). Thus a(n) <= 5 for all n. - Tom Edgar, Jan 06 2015

Examples

			For n=2, 2^k has no digit in common with 2 until k reaches 5 to give 32, hence a(2)=5.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. sequences where a(n)=k: A103173 (k=5), A189056 (k=2), A253601 (k=3), A253602 (k=4).

Cf. A373203.

Programs

Maple

f:= proc(n) local L,k;
 L:= convert(convert(n,base,10),set);
 for k from 2 do
   if convert(convert(n^k,base,10),set) intersect L <> {} then
     return k
   fi
 od
end proc:
map(f, [$0..100]); # Robert Israel, Mar 17 2020

Mathematica

seq={};Do[k=1;Until[ContainsAny[IntegerDigits[n],IntegerDigits[n^k]],k++];AppendTo[seq,k] ,{n,0,86}];seq (* James C. McMahon, Jun 04 2024 *)

PARI

a(n) = {sd = Set(vecsort(digits(n))); k=2; while (#setintersect(sd, Set(vecsort(digits(n^k)))) == 0, k++); k;}