A108859 Numbers k such that k divides the sum of the digits of k^(2k).

1, 3, 5, 9, 18, 63, 72, 74, 104, 111, 116, 117, 565, 621, 734, 1242, 1620, 4596, 4728, 5823, 5956, 21135, 28251, 46530, 46908, 78257, 129619, 277407, 463689, 464706, 599119

Offset: 1

Author

Ryan Propper, Jul 11 2005

Keywords

• base
• hard
• more
• nonn

Comments

The quotients are 1, 6, 8, 9, 13, 17, 16, 17, 17, 18, 17, 19, 25, 25, 26, 28, 20, 33, 33, 34, 34, 39, 40, 33, 42, 44, 46, 49.

Examples

			734 is a term because the sum of the digits of 734^(2*734), 19084, is divisible by 734.
		

Crossrefs

Cf. A062206.

Programs

Mathematica

Do[If[Mod[Plus @@ IntegerDigits[n^(2*n)], n] == 0, Print[n]], {n, 1, 10000}]

Python

from gmpy2 import digits, mpz
def ok(n): return n and sum(map(mpz, digits(n**(2*n))))%n == 0
print([k for k in range(2000) if ok(k)]) # Michael S. Branicky, May 08 2025

Extensions

a(22)-a(28) from Lars Blomberg, Jul 12 2011

a(29)-a(31) from Michael S. Branicky, May 13 2025