A326933 - OEIS

%I #13 Mar 14 2023 18:31:36

%S 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,

%T 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,

%U 11,1,5,5,7,1,19,1,3,8,5,3,11,1,9,8,3,1

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

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

%H Antti Karttunen, <a href="/A326933/b326933.txt">Table of n, a(n) for n = 0..2048</a>

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

%t g[x_, n_] := Numerator[ Factor[D[1/(x^2 - x + 1), {x, n}]]];

%t Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)

%t h[n_] := CoefficientList[g[x, n]/n!, x]

%t Table[h[n], {n, 0, 10}]

%t Column[%]  (* A326926 array *)

%t Table[-1 + Length[FactorList[g[x, n]/n!]], {n, 0, 100}]  (* A326933 *)

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

%Y Cf. A326926.

%K nonn

%O 0,3

%A _Clark Kimberling_, Nov 01 2019

%E Starting offset corrected from 1 to 0 by _Antti Karttunen_, Mar 02 2023