A224418 Least prime q such that sum_{k=0}^n p(k)*x^{n-k} is irreducible modulo q, where p(k) refers to the partition number A000041(k).

2, 3, 2, 11, 2, 13, 19, 19, 13, 29, 73, 47, 19, 43, 7, 59, 13, 29, 3, 13, 179, 29, 173, 19, 3, 163, 23, 3, 101, 71, 131, 977, 5, 157, 43, 13, 73, 2, 89, 197, 151, 151, 313, 3, 13, 31, 23, 97, 173, 241, 181, 109, 487, 157, 17, 29, 89, 109, 257, 317

Offset: 1

Author

Zhi-Wei Sun, Apr 06 2013

Keywords

• nonn

Comments

Conjecture: a(n) < n^2 for all n > 1.

Examples

			a(2) = 3 since sum_{k=0}^2 p(k)*x^{n-k} = x^2 + x + 2  is irreducible modulo 3 but reducible modulo 2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A000040, A000041, A224417, A224416, A220072, A223934, A224210, A217788, A224197.

Programs

Mathematica

A[n_,x_]:=A[n,x]=Sum[PartitionsP[k]*x^(n-k),{k,0,n}]
Do[Do[If[IrreduciblePolynomialQ[A[n,x],Modulus->Prime[k]]==True,Print[n," ",Prime[k]];Goto[aa]],{k,1,PrimePi[Max[1,n^2-1]]}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]