A391120 a(n) is the least k for which n^k contains at least one decimal digit 9, or -1 if there is no such k.

-1, 12, 2, 6, 8, 4, 2, 4, 1, -1, 7, 6, 2, 2, 5, 3, 2, 4, 1, 12, 3, 6, 2, 5, 4, 4, 2, 3, 1, 2, 2, 7, 2, 3, 7, 2, 2, 5, 1, 6, 3, 4, 2, 2, 3, 3, 2, 3, 1, 8, 6, 6, 2, 2, 4, 4, 2, 3, 1, 4, 3, 5, 2, 2, 5, 3, 2, 5, 1, 2, 3, 5, 2, 4, 6, 3, 2, 6, 1, 4, 6, 5, 2, 3, 6, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,2

Comments

Conjecture: a(n) <= 12, with a(n) = -1 iff n is in A011557 and a(n) = 12 iff n/2 is in A011557.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(10*k) = a(k).

Example

a(4) = 6 because 4^6 = 4096 contains the digit 9 while 4, 4^2 = 16, 4^3 = 64, 4^4 = 256 and 4^5 = 1024 do not.

Maple

f:= proc(n) local k;
  for k from 1 do if member(9, convert(n^k, base, 10)) then return k fi od
end proc:
for i from 0 to 2 do f(10^i):= -1 od:
map(f, [$1..100]);

Mathematica

a[n_]:=Module[{k=1}, If[IntegerQ[Log[10, n]], -1, While[!ContainsAny[IntegerDigits[n^k], {9}], k++]; k]]; Array[a, 98] (* James C. McMahon, Dec 04 2025 *)

Crossrefs

Cf. A011539, A011557.

Sequence in context: A054383 A036383 A107832 * A322521 A099136 A215416

Adjacent sequences: A391117 A391118 A391119 * A391121 A391122 A391123

Keyword

sign,base

Author

Robert Israel, Nov 29 2025

Status

approved