A335379 a(n) is the number of Mersenne prime (irreducible) polynomials M = x^k(x+1)^(n-k)+1 of degree n in GF(2)[x] (k goes from 1 to n-1) such that Phi_7(M) has an odd number of prime divisors (omega(Phi_7(M)) is odd).

1, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 2

Author

Luis H. Gallardo, Jun 03 2020

Keywords

• nonn

Comments

Phi_7(x)=1+x+x^2+x^3+x^4+x^5+x^6, is the 7th cyclotomic polynomial; omega(P(x)) counts the 2 X 2 distinct irreducible divisors of the binary polynomial P(x) in GF(2)[x].

It is surprising that a(n) be so small (conjecturally it is always 1 or 2). The sequence appeared when working the special case p=7 of a conjecture (see Links) about prime divisors in GF(2)[x] of the composed cyclotomic polynomial Phi_p(M), where p is a prime number and M is a Mersenne irreducible polynomial.

Examples

			For n=4 a(4)= 0 (the sequence begins a(2)=1,a(3)=2,...), since there is no Mersenne polynomial M of degree 4 in GF(2)[x] such that omega(Phi_7(M)) is odd.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..1700
• L. H. Gallardo and O. Rahavandrainy, On Mersenne polynomials over GF(2), Finite Fields Appl. 59 (2019), 284-296.

Crossrefs

Cf. A057460 (A074710), A071642, A267918, A272486, A162570.

Programs

PARI

a(n)={my(phi7=polcyclo(7)); sum(k=1, n-1, my(p=Mod(x^k * (x+1)^(n-k) + 1, 2)); polisirreducible(p) && #(factor(subst(phi7, x, p))~)%2)} \\ Andrew Howroyd, Jun 04 2020

Extensions

Terms a(22) and beyond from Andrew Howroyd, Jun 04 2020