A056739 Numbers k such that k | 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

1, 5, 11, 25, 55, 121, 125, 275, 365, 605, 625, 925, 1331, 1375, 2365, 3025, 3125, 6655, 6875, 14641, 15125, 15625, 22625, 27565, 32125, 33275, 34375, 73205, 75625, 78125, 123365, 161051, 166375, 171875, 366025, 378125, 390625, 541717, 660605

Offset: 1

Author

Robert G. Wilson v, Aug 25 2000

Keywords

• nonn

Comments

Contains A003598. In general n=p^i * q^j => n | Sum_{k=1..2*p} k^n, where p and q=2*p+1 are prime (see Meyer ref).

All terms == 1 or 5 (mod 6). The only prime terms are 5 and 11. - Robert Israel, Jun 25 2025

Links

• Robert Israel, Table of n, a(n) for n = 1..140
• Christian Meyer, On conjecture no. 22 arising from the OEIS

Crossrefs

Cf. A001557, A003598, A057291, A057292.

Programs

Maple

filter:= n ->   10 &^n + 9 &^ n + 8 &^ n + 7 &^ n + 6&^ n + 5&^n + 4&^n + 3&^n + 2&^n + 1 mod n = 0:
select(filter, [seq(seq(6*i + j, j=[1,5]),i=0..10^6)]); # Robert Israel, Jun 25 2025

Mathematica

Do[ If[ Mod[ PowerMod[ 10, n, n ] + PowerMod[ 9, n, n ] + PowerMod[ 8, n, n ] + PowerMod[ 7, n, n ] + PowerMod[ 6, n, n ] + PowerMod[ 5, n, n ] + PowerMod[ 4, n, n ] + PowerMod[ 3, n, n ] + PowerMod[ 2, n, n ] + 1, n ] == 0, Print[ n ] ], {n, 1, 10^6} ]
Select[Range[700000],Divisible[Total[Range[10]^#],#]&] (* Harvey P. Dale, Nov 24 2014 *)
Select[Range[700000],Mod[Total[PowerMod[Range[10],#,#]],#]==0&] (* Harvey P. Dale, Feb 23 2023 *)