A250198 Numbers k such that the right Aurifeuillian primitive part of 2^k+1 is prime.

2, 6, 10, 14, 18, 22, 30, 34, 38, 42, 54, 58, 66, 70, 90, 102, 110, 114, 126, 138, 170, 178, 242, 294, 314, 326, 350, 378, 462, 566, 646, 726, 758, 1150, 1242, 1302, 1482, 1558, 1638, 1710, 1770, 1970, 1994

Offset: 1

Author

Eric Chen, Jan 18 2015

Keywords

• nonn

Comments

All terms are congruent to 2 modulo 4.

Let Phi_n(x) denote the n-th cyclotomic polynomial.

Numbers n such that Phi_{2nM(n)}(2) is prime.

Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, and this is Phi_{2n}(2).

Let M(n) = the Aurifeuillian M-part of 2^n+1, M(n) = 2^(n/2) + 2^((n+2)/4) + 1 for n congruent to 2 (mod 4).

Let M*(n) = GCD(M(n), J*(n)), this sequence lists all n such that M*(n) is prime.

Examples

			14 is in this sequence because the right Aurifeuillian primitive part of 2^14+1 is 29, which is prime.
26 is not in this sequence because the right Aurifeuillian primitive part of 2^26+1 is 8321, which equals 53 * 157 and is not prime.
		

Links

• Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
• Samuel Wagstaff, The Cunningham project
• Eric W. Weisstein's World of Mathematics, Aurifeuillean Factorization.

Crossrefs

Cf. A250197, A153443, A019320, A072226, A161508, A085601, A229768, A061443.

Programs

Mathematica

Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) + 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]

PARI

isok(n) = isprime(gcd(2^(n/2) + 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ Michel Marcus, Jan 27 2015