A088817 Numbers k such that Cyclotomic(2k,k) is prime.

1, 2, 3, 4, 5, 9, 17, 36, 157, 245, 352, 3977

Offset: 1

Author

T. D. Noe, Oct 20 2003

Keywords

• hard
• nonn

Comments

This is a generalization of A056826. Note that (n^n+1)/(n+1) = cyclotomic(2n,n) when n is prime. These are probable primes for n > 352. No others < 4700.

All terms of this sequence that are greater than 3 are congruent to 0 or 1 mod 4. In general, if k = s^2*t where t is squarefree and t == 2, 3 (mod 4), then Cyclotomic(2k,t*x^2) is the product of two polynomials. See the Wikipedia link below. - Jianing Song, Sep 25 2019

Links

• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
• Wikipedia, Aurifeuillean factorization

Crossrefs

Cf. A056826 ((k^k+1)/(k+1) is prime), A070519 (cyclotomic(k,k) is prime), A088875 (cyclotomic(k,-k) is prime).

Programs

Mathematica

Do[p=Prime[n]; If[PrimeQ[Cyclotomic[2n, n]], Print[p]], {n, 100}]

PARI

is(n)=ispseudoprime(polcyclo(2*n,n)) \\ Charles R Greathouse IV, May 22 2017