A087481 Number of polynomials of the form x^n +- x^(n-1) +- x^(n-2) +- ... +- 1 irreducible over the integers.

2, 4, 4, 16, 12, 48, 64, 192, 260, 1024, 1128, 4096, 4480, 13310, 20620, 65434, 76376, 262144, 358532, 932134, 1391720, 4194090, 5447256, 16570740, 23153832, 61696126, 97361128

Offset

1,1

Comments

For each n, there are 2^n polynomials to consider. All 2^n polynomials are irreducible for n = 1, 2, 4, 10, 12, 18, which is sequence A071642. For those values of n, n+1 is a prime in Artin's primitive root conjecture (A001122).

Since p(x) is irreducible iff (-1)^n*p(-x) is irreducible, all terms are even. - Robert Israel, Dec 22 2014

Links

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

Eric Weisstein's World of Mathematics, Irreducible Polynomial

Math Overflow, Irreducible polynomials with constrained coefficients

Formula

a(n) = 2^n for n a term of A071642; see first comment.

Maple

f:= proc(n) local t, j, p0, p;
   p0:= add(x^j, j = 0 .. n);
   2*nops(select(s -> irreduc(p0 - 2*add(x^(j-1), j = s)), combinat:-powerset(n-1)));
end proc:
seq(f(n), n=1..18); # Robert Israel, Dec 22 2014

Mathematica

Irreducible[p_, n_] := Module[{f}, f=FactorList[p, Modulus->n]; Length[f]==1 || Simplify[p-f[[2, 1]]]===0]; Table[xx=x^Range[0, n-1]; cnt=0; Do[p=x^n+xx.(2*IntegerDigits[i, 2, n]-1); If[Irreducible[p, 0], cnt++ ], {i, 0, 2^n-1}]; cnt, {n, 18}]

Prog

(SageMath) R.<x>=Z[]; a(n) = sum((x^n + sum(( 2 * ((b & (1<<d)) >> d) - 1 ) * x^d for d in range(n))).is_irreducible() for b in range(2^n))

Crossrefs

Cf. A001122, A071642, A087482 (irreducible binary polynomials).

Sequence in context: A246047 A079102 A071337 * A038210 A244640 A230874

Adjacent sequences: A087478 A087479 A087480 * A087482 A087483 A087484

Keyword

nonn,hard,more

Author

T. D. Noe, Sep 09 2003

Extensions

a(19) from Robert Israel, Dec 22 2014

a(20)-a(27) from Lucas A. Brown, May 19 2023

Status

approved