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%I #43 Nov 23 2019 04:08:35
%S 6,9,12,14,17,23,44,47,63,84,129,236,278,279,297,647,726,737,2574,
%T 4233,8207,16046,21983,23999,24596,24849,84929
%N Numbers n such that x^(n-5)*(x+1)^5+1 is irreducible in F2[x].
%C 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.
%C 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).
%C Next term > 10^5. - _Joerg Arndt_, May 01 2016
%H E. F. Canaday, <a href="http://dx.doi.org/10.1215/S0012-7094-41-00861-X">The sum of the divisors of a polynomial</a>, Duke Math. J. 8, (1941), 721-737.
%e For n=6, x^(6-5)*(x+1)^5+1 = x^6 + x^5 + x^2 + x + 1 is irreducible in F_2[x].
%o (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
%o (Sage)
%o P.<x> = GF(2)[]
%o for n in range(6, 10^5):
%o if (x^(n-5)*(1+x)^5+1).is_irreducible():
%o print(n)
%o # _Joerg Arndt_, May 01 2016
%K nonn,more
%O 1,1
%A _Luis H. Gallardo_, May 01 2016
%E Terms a(12) and beyond from _Joerg Arndt_, May 01 2016
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