A335383 a(n) is the number of irreducible Mersenne polynomials in GF(2)[x] that have degree n.

1, 2, 2, 2, 2, 4, 0, 4, 2, 2, 2, 0, 2, 6, 0, 6, 2, 0, 2, 2, 2, 4, 0, 4, 0, 0, 8, 2, 2, 8, 0, 4, 2, 2, 2, 0, 0, 6, 0, 4, 0, 0, 2, 0, 2, 8, 0, 8, 0, 0, 8, 0, 0, 4, 0, 8, 2, 0, 8, 0, 2, 8, 0, 4, 0, 0, 4, 0, 0, 10, 0, 6, 2, 0, 2, 0, 0, 4, 0, 6, 0, 0, 6, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 4, 0, 8, 4, 0, 6

Offset: 2

Author

Luis H. Gallardo, Jun 04 2020

Keywords

• nonn

Comments

A Mersenne polynomial is a binary (i.e., an element of GF(2)[x]) polynomial M, of degree > 1, such that M+1 has only 0 and 1 as roots in a fixed algebraic closure of GF(2).

If for some positive integers a,b, M = x^a(x+1)^b+1 is an irreducible Mersenne polynomial, then gcd(a,b)=1. This condition is not sufficient.

There is no known formula for a(n). Of course it is bounded above by the total number of prime (irreducible) binary polynomials of degree n, but this is a too weak upper bound. A trivial, better upper bound, is simply n-1, the total number of Mersenne polynomials (prime or not) of degree n.

Examples

			For n = 5 one has a(5) = 2 since there are 2 irreducible Mersenne polynomials of degree 5. Namely, x^2*(x+1)^3+1 and x^3*(x+1)^2+1.
For n = 8, a(8) = 0 since there are no irreducible Mersenne polynomial of degree 8.
		

Links

• L. H. Gallardo and O. Rahavandrainy, On even (unitary) perfect polynomials over F_2, Finite Fields Appl. 18, no. 5, (2012), 920-932.
• L. H. Gallardo and O. Rahavandrainy, Characterization of Sporadic perfect polynomials over F_2, Funct. Approx. Comment. Math.,(2016), 7-21.
• L. H. Gallardo and O. Rahavandrainy, On Mersenne polynomials over F_2, Finite Fields Appl. 59 (2019), 284-296.
• L. H. Gallardo and O. Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.

Crossrefs

Cf. A267918, A272486, A162570.

Programs

PARI

a(n) = sum(k=1, n-1, polisirreducible(Mod(1, 2)*(x^(n-k)*(x+1)^k+1))); \\ Michel Marcus, Jun 07 2020

Formula

a(A272486(n)) = 0. - Michel Marcus, Jun 07 2020