A326933 Number of nonconstant irreducible polynomial divisors of the n-th polynomial given in A326926.

0, 1, 2, 2, 1, 5, 1, 3, 4, 3, 1, 8, 1, 3, 5, 4, 1, 9, 1, 5, 5, 3, 1, 11, 2, 3, 6, 5, 1, 11, 1, 5, 5, 3, 3, 14, 1, 3, 5, 7, 1, 11, 1, 5, 9, 3, 1, 14, 2, 5, 5, 5, 1, 13, 3, 7, 5, 3, 1, 17, 1, 3, 9, 6, 3, 11, 1, 5, 5, 7, 1, 19, 1, 3, 8, 5, 3, 11, 1, 9, 8, 3, 1

Offset: 0

Author

Clark Kimberling, Nov 01 2019

Keywords

• nonn

Comments

It appears that each nonconstant polynomial is irreducible if and only if its degree is p-1 for some prime p other than 3.

Examples

			The 5 nonconstant irreducible divisors of the 5th polynomial appear in this factorization: -3 x (-2 + x) (-1 + x) (1 + x) (-1 + 2 x).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..2048

Crossrefs

Cf. A326926.

Programs

Mathematica

g[x_, n_] := Numerator[ Factor[D[1/(x^2 - x + 1), {x, n}]]];
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x]
Table[h[n], {n, 0, 10}]
Column[%]  (* A326926 array *)
Table[-1 + Length[FactorList[g[x, n]/n!]], {n, 0, 100}]  (* A326933 *)

PARI

A326933(n) = { my(p=1/(1-x+x^2)); for(k=1,n, p = deriv(p)); #(factor(numerator(p)/n!)~); };

Extensions

Starting offset corrected from 1 to 0 by Antti Karttunen, Mar 02 2023