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Numbers n such that the left Aurifeuillian primitive part of 2^n+1 is prime.
3

%I #72 Jun 23 2019 17:00:23

%S 10,14,18,22,26,30,42,54,58,66,70,86,94,98,106,110,126,130,138,146,

%T 158,174,186,210,222,226,258,302,334,434,462,478,482,522,566,602,638,

%U 706,734,750,770,782,914,1062,1086,1114,1126,1226,1266,1358,1382,1434,1742,1926

%N Numbers n such that the left Aurifeuillian primitive part of 2^n+1 is prime.

%C All terms are congruent to 2 modulo 4.

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

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

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

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

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

%H Eric Chen, Gord Palameta, <a href="/A250197/a250197_2.txt">Factorization of Phi_n(2) for n up to 1280</a>

%H Samuel Wagstaff, <a href="http://homes.cerias.purdue.edu/~ssw/cun/index.html">The Cunningham project</a>

%H Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AurifeuilleanFactorization.html">Aurifeuillean Factorization</a>.

%e 14 is in this sequence because the left Aurifeuillian primitive part of 2^14+1 is 113, which is prime.

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

%o (PARI) isok(n) = isprime(gcd(2^(n/2) - 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ _Michel Marcus_, Jan 27 2015

%Y Cf. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.

%K nonn

%O 1,1

%A _Eric Chen_, Jan 18 2015