

A250197


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


3



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
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OFFSET

1,1


COMMENTS

All terms are congruent to 2 modulo 4.
Phi_n(x) is the nth 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 Lpart 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.


LINKS

Table of n, a(n) for n=1..54.
Eric Chen, Factorization of Phi_n(2) for n up to 1280
Samuel Wagstaff, The Cunningham project
Eric W. Weisstein's World of Mathematics, Aurifeuillean Factorization.


EXAMPLE

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.


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.
Sequence in context: A096851 A244033 A121893 * A055985 A190888 A157138
Adjacent sequences: A250194 A250195 A250196 * A250198 A250199 A250200


KEYWORD

nonn


AUTHOR

Eric Chen, Jan 18 2015


STATUS

approved



