A275528 Prime factors of numbers of the form 4^(2^m) - 2^(2^m) + 1 with m >= 0.

3, 13, 97, 193, 241, 673, 769, 12289, 786433, 22253377, 39714817, 152371201, 597688321, 1107296257, 3221225473, 7348420609, 11560943617, 29796335617, 74490839041, 77309411329, 206158430209, 246423748609, 448203325441, 2422022479873, 5469640851457, 28114855919617

Offset

1,1

Comments

Primes p other than 3 such that one third of the multiplicative order of 2 (mod p) is a power of 2.

Primes in A255772 (except 7), sorted.

Links

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..27

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.

Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.

Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.

Example

3 divides 2^2 - 2^1 + 1 = 3.
13 divides 2^4 - 2^2 + 1 = 13.
97 divides 2^16 - 2^8 + 1 = 65281.
193 divides 2^32 - 2^16 + 1 = 4294901761.
241 divides 2^8 - 2^4 + 1 = 241.
673 divides 2^16 - 2^8 + 1 = 65281.
769 divides 2^128 - 2^64 + 1 = 340282366920938463444927863358058659841.
12289 divides 2^2048 - 2^1024 + 1.

Prog

(PARI) forprime(p=3, 10^15, o=znorder(Mod(2, p))/3; x=ispower(2*o); if(p==3||2^(x-1)==o, print1(p, ", ")));

Crossrefs

Cf. A220294, A255772.

Sequence in context: A006898 A277492 A342305 * A129375 A293528 A228160

Adjacent sequences: A275525 A275526 A275527 * A275529 A275530 A275531

Keyword

nonn

Author

Arkadiusz Wesolowski, Jul 31 2016

Status

approved