A272486 -id:A272486 - cp's OEIS frontend

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

Luis H. Gallardo, Jun 03 2020

nonn

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.

			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.

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.

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

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

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

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

Original entry on oeis.org

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

Luis H. Gallardo, Jun 04 2020

nonn

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.

			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.

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.

Cf. A267918, A272486, A162570.

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

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

cp's OEIS Frontend

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).

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