A394058 Conjectured least integer k>1 such that n^k has the smallest average of digits, or -1 if no such k exists.

2, 10, 21, 5, 13, 3, 4, 3, 6, -1, 2, 2, 44, 7, 2, 25, 20, 7, 4, 10, 5, 12, 12, 8, 6, 7, 7, 5, 10, 21, 7, 2, 8, 7, 2, 11, 25, 2, 2, 5, 6, 5, 16, 8, 2, 2, 3, 2, 2, 13, 2, 5, 6, 11, 2, 2, 4, 3, 10, 3, 2, 7, 3, 3, 5, 11, 4, 3, 4, 4, 2, 7, 12, 7, 8, 15, 4, 4, 2, 3

Offset

1,1

Comments

All terms have been verified up to k <= 20000.

For n <= 84000, the largest term is a(38541) = 188.

Links

James C. McMahon, Table of n, a(n) for n = 1..5000

Example

a(3) = 21, because 3^21 = 10460353203, which has the smallest average of digits of ~2.45 for any 3^k and 1 < k <= 20000.
a(10) = -1, because the average of digits of 10^k decreases indefinitely as k increases.

Mathematica

a[1]=2; a[n_]:=Module[{s=DigitSum[#]/IntegerLength[#]&/@(n^Range[2, 200])}, If[IntegerQ[Log10[n]], -2, SequencePosition[s, {Min[s]}][[1, 1]]]+1] (* Increase range of k for n^k when n>84000 *); Array[a, 80] (* James C. McMahon, Mar 21 2026 *)

Crossrefs

Cf. A394057.

Sequence in context: A284213 A156818 A037422 * A136735 A294549 A294550

Adjacent sequences: A394055 A394056 A394057 * A394059 A394060 A394061

Keyword

sign,base

Author

Dmitry Kamenetsky, Mar 09 2026

Status

approved