A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

10, 14, 18, 22, 26, 30, 42, 54, 58, 66, 70, 86, 94, 98, 106, 110, 126, 130, 138, 146, 158, 174, 186, 210, 222, 226, 258, 302, 334, 434, 462, 478, 482, 522, 566, 602, 638, 706, 734, 750, 770, 782, 914, 1062, 1086, 1114, 1126, 1226, 1266, 1358, 1382, 1434, 1742, 1926

Offset: 1

Author

Eric Chen, Jan 18 2015

Keywords

• nonn

Comments

All terms are congruent to 2 modulo 4.

Phi_n(x) is the n-th cyclotomic polynomial.

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

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

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

Let L*(n) = GCD(L(n), J*(n)).

This sequence lists all n such that L*(n) is prime.

Examples

			14 is in this sequence because the left Aurifeuillian primitive part of 2^14+1 is 113, which is prime.
34 is not in this sequence because the left Aurifeuillian primitive part of 2^34+1 is 130561, which equals 137 * 953 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. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.

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