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A087481 Number of polynomials of the form x^n +- x^(n-1) +- x^(n-2) +- ... +- 1 irreducible over the integers. 2
2, 4, 4, 16, 12, 48, 64, 192, 260, 1024, 1128, 4096, 4480, 13310, 20620, 65434, 76376, 262144, 358532 (list; graph; refs; listen; history; text; internal format)
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..19.

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}]

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,more

AUTHOR

T. D. Noe, Sep 09 2003

EXTENSIONS

a(19) from Robert Israel, Dec 22 2014

STATUS

approved

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Last modified January 24 12:29 EST 2022. Contains 350537 sequences. (Running on oeis4.)