A239134 Smallest k such that n^k contains k as a substring in its decimal representation.

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, 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, 3, 3, 1, 3, 2, 6, 2, 3, 2, 6, 2, 4, 1, 2, 4, 4

Offset: 1

Author

Derek Orr, Mar 10 2014

Keywords

• nonn
• base

Comments

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

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

Counterexample: a(10^d - 2) = 6 for d >= 2. - Robert Israel, Sep 16 2024

Examples

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A045537, A061280.

Programs

Maple

f:= proc(n) local k;
  for k from 1 to 9 do
    if member(k,convert(n^k,base,10)) then return k fi
  od;
  FAIL
end proc:
map(f, [$1..100]); # Robert Israel, Sep 16 2024

Mathematica

a[n_] := Block[{k=1}, While[{} == StringPosition[ ToString[n^k], ToString[k]], k++]; k]; Array[a, 84] (* Giovanni Resta, Mar 11 2014 *)
sk[n_]:=Module[{k=1},While[SequenceCount[IntegerDigits[n^k],IntegerDigits[k]] == 0,k++];k]; Array[sk,90] (* Harvey P. Dale, May 12 2022 *)

Python

def Sub(x):
  for n in range(10**3):
    if str(x**n).find(str(n)) > -1:
      return n
x = 1
while x < 10**3:
  print(Sub(x))
  x += 1

Formula

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

a(10*n) = a(n) if a(n) < 10. - Robert Israel, Sep 16 2024