A072226 Numbers k such that the k-th cyclotomic polynomial evaluated at 2 (=A019320(k)) is prime.

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261

Offset

1,1

Comments

The prime n in this sequence are in A000043, which produce the Mersenne primes. If 2p is in this sequence, with p prime, then p is a Wagstaff number, A000978. - T. D. Noe, Apr 02 2008

While the sequence looks quite dense for small values, note that there are only 10 values in the interval [700,1200]. - M. F. Hasler, Apr 03 2008

No term greater than 12 can be congruent to 4 modulo 8 as proved by Schinzel (1962), see also Pomerance (2024). Note the Aurifeuillean factorization: Product_{4|d, d|8*k+4} Phi(d,2) = 2^(4k+2) + 1 = (2^(2k+1) - 2^(k+1) + 1) * (2^(2k+1) + 2^(k+1) + 1). If Phi(8*k+4,2) is prime, then it divides either 2^(2k+1) - 2^(k+1) + 1 or 2^(2k+1) + 2^(k+1) + 1. This immediately proves that no term can be of the form 4*p for odd primes p >= 5 Since Phi(4*p,2) = (2^(2*p) + 1)/5. - Jianing Song, Apr 04 2022; edited by Max Alekseyev, Dec 03 2024

Links

Max Alekseyev, Table of n, a(n) for n = 1..289 (terms 1..234, 235..277, and 278..289 from Yves Gallot, T. D. Noe, and Carl Pomerance, respectively)

Joerg Arndt, Matters Computational (The Fxtbook)

Yves Gallot, Cyclotomic polynomials and prime numbers

Carl Pomerance, Cyclotomic primes, arXiv:2411.04213 [math.NT], 2024.

Wikipedia, Aurifeuillean factorization

Index entries for cyclotomic polynomials, values at X

Mathematica

Select[Range[600], PrimeQ[Cyclotomic[ #, 2]]&]

Prog

(PARI) for( i=1, 999, ispseudoprime( polcyclo(i, 2)) && print1( i", ")) /* for PARI < 2.4.2 use ...subst(polcyclo(i), x, 2)... */ \\ M. F. Hasler, Apr 03 2008

Crossrefs

Corresponding primes are listed in A292015.

Cf. A138920-A138940.

Sequence in context: A004744 A171987 A332111 * A247809 A247802 A245028

Adjacent sequences: A072223 A072224 A072225 * A072227 A072228 A072229

Keyword

nonn

Author

Reiner Martin, Jul 04 2002

Extensions

Edited by Max Alekseyev, Apr 25 2018

Status

approved