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

Author

Reiner Martin, Jul 04 2002

Keywords

• nonn

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

Crossrefs

Corresponding primes are listed in A292015.

Cf. A138920-A138940.

Programs

Mathematica

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

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

Extensions

Edited by Max Alekseyev, Apr 25 2018