cp's OEIS Frontend

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

%I A250197 #79 Jun 30 2025 14:38:28
%S A250197 10,14,18,22,26,30,42,54,58,66,70,86,94,98,106,110,126,130,138,146,
%T A250197 158,174,186,210,222,226,258,302,334,434,462,478,482,522,566,602,638,
%U A250197 706,734,750,770,782,914,1062,1086,1114,1126,1226,1266,1358,1382,1434,1742,1926
%N A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.
%C A250197 All terms are congruent to 2 modulo 4.
%C A250197 Phi_n(x) is the n-th cyclotomic polynomial.
%C A250197 Numbers n such that Phi_{2nL(n)}(2) is prime.
%C A250197 Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, this is Phi_{2n}(2).
%C A250197 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).
%C A250197 Let L*(n) = GCD(L(n), J*(n)).
%C A250197 This sequence lists all n such that L*(n) is prime.
%H A250197 Eric Chen, Gord Palameta, <a href="/A250197/a250197_2.txt">Factorization of Phi_n(2) for n up to 1280</a>
%H A250197 Samuel Wagstaff, <a href="http://homes.cerias.purdue.edu/~ssw/cun/index.html">The Cunningham project</a>
%H A250197 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AurifeuilleanFactorization.html">Aurifeuillean Factorization</a>.
%e A250197 14 is in this sequence because the left Aurifeuillian primitive part of 2^14+1 is 113, which is prime.
%e A250197 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.
%t A250197 Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) - 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]
%o A250197 (PARI) isok(n) = isprime(gcd(2^(n/2) - 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ _Michel Marcus_, Jan 27 2015
%Y A250197 Cf. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.
%K A250197 nonn
%O A250197 1,1
%A A250197 _Eric Chen_, Jan 18 2015