A353159 Integers k for which there exists some integer m such that the sum of the digits of m^k is equal to m + k.

2, 3, 6, 9, 12, 15, 18, 20, 21, 24, 27, 30, 33, 36, 38, 39, 42, 45, 48, 51, 54, 56, 57, 63, 66, 69, 72, 74, 75, 78, 81, 84, 87, 90, 92, 93, 96, 99, 102, 105, 108, 110, 111, 114, 117, 120, 123, 126, 129, 132, 135, 141, 144, 146, 147, 150, 153, 156, 159, 162

Offset: 1

Author

Samuel Owen, Apr 27 2022

Keywords

• nonn
• base

Comments

Letting t = m^k, this sequence consists of the integers k for which there exists some integer m such that s(t) = m + k, where s(t) = A007953(t) represents the sum of digits of t. Rearranging gives m = t^(1/k) = s(t) - k; this allows you to find numbers which follow a common online trick like 64^(1/2) = (6 + 4) - 2 or 216^(1/3) = (2 + 1 + 6) - 3. This online trick was the original motivation for this sequence.

Examples

			s(62^9) = 62 + 9, so 9 is a term.
s(2157^156) = 2157 + 156, so 156 is a term.
s(18045^999) = 18045 + 999,  so 999 is a term.
		

Links

• Samuel Owen, Table of n, a(n) for n = 1..366
• Samuel Owen, Every value of m, for each integer k for k = 2..999

Crossrefs

Cf. A007953, A133509.

Formula

For any given k, the value of m is bounded by 0 < m < x, where x is the maximum solution to the equation x = 10^(1/k)*k*floor(9*log_10(x)-1).