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

%I A250198 #73 Jun 30 2025 14:38:42
%S A250198 2,6,10,14,18,22,30,34,38,42,54,58,66,70,90,102,110,114,126,138,170,
%T A250198 178,242,294,314,326,350,378,462,566,646,726,758,1150,1242,1302,1482,
%U A250198 1558,1638,1710,1770,1970,1994
%N A250198 Numbers k such that the right Aurifeuillian primitive part of 2^k+1 is prime.
%C A250198 All terms are congruent to 2 modulo 4.
%C A250198 Let Phi_n(x) denote the n-th cyclotomic polynomial.
%C A250198 Numbers n such that Phi_{2nM(n)}(2) is prime.
%C A250198 Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, and this is Phi_{2n}(2).
%C A250198 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).
%C A250198 Let M*(n) = GCD(M(n), J*(n)), this sequence lists all n such that M*(n) is prime.
%H A250198 Eric Chen, Gord Palameta, <a href="https://oeis.org/A250197/a250197_2.txt">Factorization of Phi_n(2) for n up to 1280</a>
%H A250198 Samuel Wagstaff, <a href="http://homes.cerias.purdue.edu/~ssw/cun/index.html">The Cunningham project</a>
%H A250198 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AurifeuilleanFactorization.html">Aurifeuillean Factorization</a>.
%e A250198 14 is in this sequence because the right Aurifeuillian primitive part of 2^14+1 is 29, which is prime.
%e A250198 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.
%t A250198 Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) + 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]
%o A250198 (PARI) isok(n) = isprime(gcd(2^(n/2) + 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ _Michel Marcus_, Jan 27 2015
%Y A250198 Cf. A250197, A153443, A019320, A072226, A161508, A085601, A229768, A061443.
%K A250198 nonn
%O A250198 1,1
%A A250198 _Eric Chen_, Jan 18 2015