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A267918 Numbers n such that x^(n-5)*(x+1)^5+1 is irreducible in F2[x]. 2
6, 9, 12, 14, 17, 23, 44, 47, 63, 84, 129, 236, 278, 279, 297, 647, 726, 737, 2574, 4233, 8207, 16046, 21983, 23999, 24596, 24849, 84929 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..27.

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

Sequence in context: A072546 A295670 A272466 * A330703 A262828 A306647

Adjacent sequences:  A267915 A267916 A267917 * A267919 A267920 A267921

KEYWORD

nonn,more

AUTHOR

Luis H. Gallardo, May 01 2016

EXTENSIONS

Terms a(12) and beyond from Joerg Arndt, May 01 2016

STATUS

approved

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Last modified August 10 14:50 EDT 2020. Contains 336381 sequences. (Running on oeis4.)