A373580 Numbers k that divide 2^(2^k) - 2^k + 1.

1, 3, 5, 17, 257, 641, 1605, 4369, 32113, 65537, 94945, 114689, 159441, 164737, 274177, 319489, 974849, 2424833, 3862465, 6700417, 13631489, 13906833, 16843009, 26017793, 42009217, 45592577, 63766529, 70463489, 167772161, 175747457, 825753601, 1214251009, 1227890731, 1251711641

Offset: 1

Author

Thomas Ordowski, Jun 10 2024

Keywords

• nonn

Comments

Numbers k that divide A119564(k).

A term is prime if and only if it is in A023394.

If k is in A307843, then it is a term of this sequence.

The terms that are not divisors of Fermat numbers are 1605, 4369, 32113, 94945, ... (they are all composite). Are there infinitely many of them?

Note that 2^(2^k) - 2^k + 1 = (2^(2^k) - 1) - (2^k - 2).

Crossrefs

Cf. A023394 (primes in this sequence), A119564, A307843 (subsequence).

Programs

Mathematica

q[k_] := Mod[PowerMod[2, 2^k, k] - PowerMod[2, k, k] + 1, k] == 0; Select[Range[1, 10^5, 2], q] (* Amiram Eldar, Jun 10 2024 *)

PARI

isok(k) = Mod(Mod(2, k)^(2^k) - Mod(2,k)^k + 1, k) == 0; \\ Michel Marcus, Jun 12 2024

Extensions

More terms from Amiram Eldar, Jun 10 2024