A087481 -id:A087481 - cp's OEIS frontend

A087610 Number of (-1,0,1) polynomials of degree-n irreducible over the integers.

3, 5, 12, 34, 104, 292, 916, 2791, 8660, 26538, 81584, 248554

Offset: 1

T. D. Noe, Sep 11 2003

A (-1,0,1) polynomial is defined as a monic polynomial whose remaining coefficients are either -1, 0, or 1. For each n, there are 3^n polynomials to consider.

			a(2) = 5 because 1+x+x^2, 1+x^2, 1-x+x^2, -1+x+x^2, -1-x+x^2 are irreducible over the integers.

Eric Weisstein's World of Mathematics, Irreducible Polynomial

Cf. A087481 (irreducible polynomials of the form x^n +- x^(n-1) +- x^(n-2) +- ... +- 1), A087482 (irreducible binary polynomials).

F:= proc(n) local T, count, t,x,p;
      if n::odd then
        T:= combinat:-cartprod([[-1,0,1]$(n-1),[1]])
      else
        T:= combinat:-cartprod([[-1,0,1]$(n-1),[-1,1]])
      fi;
      count:= 0;
      while not T[finished] do
         t:= T[nextvalue]();
         p:= x^n + add(t[i]*x^(n-i),i=1..n);
         if irreduc(p) then count:= count+1 fi;
      od;
      if n::odd then 2*count else count fi;
end proc:
3, seq(F(n),n=2..11); # Robert Israel, Dec 10 2015

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.(IntegerDigits[i, 3, n]-1); If[Irreducible[p, 0], cnt++ ], {i, 0, 3^n-1}]; cnt, {n, 10}]

a(11) and a(12) from Robert Israel, Dec 10 2015

A087482 Number of binary polynomials of degree n irreducible over the integers.

Original entry on oeis.org

2, 2, 2, 6, 8, 21, 34, 84, 150, 331, 614, 1417, 2638, 5508, 10874, 23437, 44862, 95887, 185238, 390297, 765510, 1557427, 3043918, 6525948, 12706892, 25836122, 51135384, 105070336, 206266718, 426254492

Offset: 1

T. D. Noe, Sep 09 2003

A binary polynomial is defined as a monic polynomial whose remaining coefficients are either 0 or 1. For each n, there are 2^n polynomials to consider.

Eric Weisstein's World of Mathematics, Irreducible Polynomial

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

Cf. A001037 (irreducible polynomials over GF(2)).

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.(IntegerDigits[i, 2, n]); If[Irreducible[p, 0], cnt++ ], {i, 0, 2^n-1}]; cnt, {n, 16}]

a(n)= { if( n<=2, return(2)); my(d, P, ct=0, x='x);  forstep (k=1, 2^n-1, 2, P=x^n+Pol(binary(k),x); ct+=polisirreducible(P) );  return(ct); }
for(n=1,30,print1(a(n),", ")); \\ Joerg Arndt, Dec 22 2014

a(n) >= A001037(n). - Joerg Arndt, Dec 22 2014

Added more terms, Joerg Arndt, Dec 22 2014

a(23)-a(30) from Max Alekseyev, May 07 2022

cp's OEIS Frontend

A087610 Number of (-1,0,1) polynomials of degree-n irreducible over the integers.

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