A378696 Numbers k such that omega(k)^k == omega(k) (mod k), where omega = A001221.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 66, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243

Offset: 1

Author

Juri-Stepan Gerasimov, Dec 04 2024

Keywords

• nonn

Comments

The sequence without A000961 and A001567 is 2665, 3367, 5551, 7107, 8205, 11011, 15457, 16471, 19345 ,... (see A379056).

Crossrefs

Supersequence of A000961 and A002997.

Cf. A001221, A001567, A214305, A379056.

Programs

Magma

[k: k in [1..250] | #PrimeDivisors(k)^k mod k eq #PrimeDivisors(k)];

Maple

filter:= proc(k) local w;
  w:= nops(numtheory:-factorset(k));
  w &^k - w mod k = 0
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 08 2024

Mathematica

q[k_] := Module[{om = PrimeNu[k]}, PowerMod[om, k, k] == om]; Select[Range[250], q] (* Amiram Eldar, Dec 06 2024 *)

PARI

isok(k) = my(x=omega(k)); Mod(x, k)^k == Mod(x, k); \\ Michel Marcus, Dec 04 2024