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

1,2

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

Links

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

Formula

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A045537, A061280.

Sequence in context: A321094 A330114 A011423 * A196616 A369104 A253271

Adjacent sequences: A239131 A239132 A239133 * A239135 A239136 A239137

Keyword

nonn,base

Author

Derek Orr, Mar 10 2014

Status

approved