A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

2, 3, 4, 6, 10, 12, 14, 19, 31, 46, 74, 75, 98, 102, 126, 180, 236, 310, 368, 1770, 1858, 3512, 4878, 5730, 7547, 7990, 8636, 9378, 11262

Offset

1,1

Comments

When n is prime, then the solutions are given in A088790.

No term of this sequence is congruent to 1 mod 4. In general, if k = s^2*t where t is squarefree and t == 1 (mod 4), then Cyclotomic(k,t*x^2) is the product of two polynomials. See the Wikipedia link below. - Jianing Song, Sep 25 2019

All terms <= 1858 have been proven with PARI's implementation of ECPP. All larger terms are BPSW PRPs. There are no further terms <= 30000. - Lucas A. Brown, Dec 28 2020

Links

Table of n, a(n) for n=1..29.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Wikipedia, Aurifeuillean factorization

Mathematica

Do[s=Cyclotomic[n, n]; If[PrimeQ[s], Print[n]], {n, 2, 256}]

Prog

(PARI) for(n=2, 10^9, if(ispseudoprime(polcyclo(n, n)), print1(n, ", "))); \\ Joerg Arndt, Jan 22 2015

Crossrefs

Cf. A070518, A070520, A088790 ((k^k-1)/(k-1) is prime), A088817 (cyclotomic(2k,k) is prime), A088875 (cyclotomic(k,-k) is prime).

Cf. A117544, A085398.

Sequence in context: A191149 A344552 A122907 * A177913 A098392 A076850

Adjacent sequences: A070516 A070517 A070518 * A070520 A070521 A070522

Keyword

nonn,more

Author

Labos Elemer, May 02 2002

Extensions

More terms from T. D. Noe, Oct 17 2003

a(29) from Charles R Greathouse IV, May 05 2011

Status

approved