A250197 -id:A250197 - cp's OEIS frontend

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

2, 6, 10, 14, 18, 22, 30, 34, 38, 42, 54, 58, 66, 70, 90, 102, 110, 114, 126, 138, 170, 178, 242, 294, 314, 326, 350, 378, 462, 566, 646, 726, 758, 1150, 1242, 1302, 1482, 1558, 1638, 1710, 1770, 1970, 1994

Offset: 1

Eric Chen, Jan 18 2015

nonn

All terms are congruent to 2 modulo 4.
Let Phi_n(x) denote the n-th cyclotomic polynomial.
Numbers n such that Phi_{2nM(n)}(2) is prime.
Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, and this is Phi_{2n}(2).
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).
Let M*(n) = GCD(M(n), J*(n)), this sequence lists all n such that M*(n) is prime.

			14 is in this sequence because the right Aurifeuillian primitive part of 2^14+1 is 29, which is prime.
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.

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.

Cf. A250197, A153443, A019320, A072226, A161508, A085601, A229768, A061443.

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

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

A250203 Numbers n such that the Phi_n(2) is the product of exactly two primes and is divisible by 2n+1.

Original entry on oeis.org

11, 20, 23, 35, 39, 48, 83, 96, 131, 231, 303, 375, 384, 519, 771, 848, 1400, 1983, 2280, 2640, 2715, 3359, 6144, 7736, 7911, 11079, 13224, 16664, 24263, 36168, 130439, 406583

Offset: 1

Eric Chen, Mar 13 2015

Here Phi_n is the n-th cyclotomic polynomial.
Is this sequence infinite?
Phi_n(2)/(2n+1) is only a probable prime for n > 16664.
a(33) > 2000000.
Subsequence of A005097 (2 * a(n) + 1 are all primes)
Subsequence of A081858.
2 * a(n) + 1 are in A115591.
Primes in this sequence are listed in A239638.
A085021(a(n)) = 2.
All a(n) are congruent to 0 or 3 (mod 4). (A014601)
All a(n) are congruent to 0 or 2 (mod 3). (A007494)
Except the term 20, all even numbers in this sequence are divisible by 8.

			Phi_11(2) = 23 * 89 and 23 = 2 * 11 + 1, so 11 is in this sequence.
Phi_35(2) = 71 * 122921 and 71 = 2 * 35 + 1, so 35 is in this sequence.
Phi_48(2) = 97 * 673 and 97 = 2 * 48 + 1, so 48 is in this sequence.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Will Edgington, Factorization of completely factored Phi_n(2) [from Internet Archive Wayback Machine]
Henri Lifchitz and Renaud Lifchitz, PRP records. Search for (2^a-1)/b
Samuel Wagstaff, The Cunningham project

Cf. A239638, A085724, A072226, A005384, A005097, A081858, A014601, A014664, A001917, A115591.

Select[Range[10000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
PrimeQ[Cyclotomic[#, 2]/(2*#+1)] &]

isok(n) = if (((x=polcyclo(n, 2)) % (2*n+1) == 0) && (omega(x) == 2), print1(n, ", ")); \\ Michel Marcus, Mar 13 2015

cp's OEIS Frontend

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

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