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

%I A335383 #49 Aug 26 2021 11:44:12
%S A335383 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,
%T A335383 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,
%U A335383 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
%N A335383 a(n) is the number of irreducible Mersenne polynomials in GF(2)[x] that have degree n.
%C A335383 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).
%C A335383 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.
%C A335383 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.
%H A335383 L. H. Gallardo and O. Rahavandrainy, <a href="https://doi.org/10.1016/j.ffa.2012.06.004">On even (unitary) perfect polynomials over F_2</a>, Finite Fields Appl. 18, no. 5, (2012), 920-932.
%H A335383 L. H. Gallardo and O. Rahavandrainy, <a href="https://projecteuclid.org/euclid.facm/1474301226">Characterization of Sporadic perfect polynomials over F_2</a>, Funct. Approx. Comment. Math.,(2016), 7-21.
%H A335383 L. H. Gallardo and O. Rahavandrainy, <a href="https://doi.org/10.1016/j.ffa.2019.06.006">On Mersenne polynomials over F_2</a>, Finite Fields Appl. 59 (2019), 284-296.
%H A335383 L. H. Gallardo and O. Rahavandrainy, <a href="https://arxiv.org/abs/1908.00106">On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors</a>, arXiv:1908.00106 [math.NT], 2019.
%F A335383 a(A272486(n)) = 0. - _Michel Marcus_, Jun 07 2020
%e A335383 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.
%e A335383 For n = 8, a(8) = 0 since there are no irreducible Mersenne polynomial of degree 8.
%o A335383 (PARI) a(n) = sum(k=1, n-1, polisirreducible(Mod(1, 2)*(x^(n-k)*(x+1)^k+1))); \\ _Michel Marcus_, Jun 07 2020
%Y A335383 Cf. A267918, A272486, A162570.
%K A335383 nonn
%O A335383 2,2
%A A335383 _Luis H. Gallardo_, Jun 04 2020