OFFSET
1,1
COMMENTS
Putting M(n,a) = x^(n-a)*(x+1)^n+1 in F2[x], a "Mersenne binary polynomial" and S(n,a) = x^n +(x+1)^a in F2[x], we see that the n's in the sequence are also the n's where S(n,5) is irreducible.
Irreducible Mersenne binary polynomials appear as factors of the only eleven known (see Canaday's paper) nontrivial even perfect polynomials over F2, i.e., polynomials A in F2[x], divisible by x*(x+1), that are fixed points of the sum of divisors function sigma. In other words, we also have sigma(A)=A, where sigma(A) is the sum in F2[x] of all divisors of A (including 1 and A). Trivial even perfect polynomials are the M(2^(n+1)-2,2^n-1)+1 = x^(2^n-1)*(x+1)^(2^n-1).
Next term > 10^5. - Joerg Arndt, May 01 2016
LINKS
E. F. Canaday, The sum of the divisors of a polynomial, Duke Math. J. 8, (1941), 721-737.
EXAMPLE
For n=6, x^(6-5)*(x+1)^5+1 = x^6 + x^5 + x^2 + x + 1 is irreducible in F_2[x].
PROG
(PARI) for(n=5, 10^5, if(polisirreducible(Mod(1, 2)*(x^(n-5)*(x+1)^5+1)), print1(n, ", "))); \\ Joerg Arndt, May 01 2016
(Sage)
P.<x> = GF(2)[]
for n in range(6, 10^5):
if (x^(n-5)*(1+x)^5+1).is_irreducible():
print(n)
# Joerg Arndt, May 01 2016
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Luis H. Gallardo, May 01 2016
EXTENSIONS
Terms a(12) and beyond from Joerg Arndt, May 01 2016
STATUS
approved