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

1,1

Comments

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

Links

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A078828 A247170 A241055 * A220947 A224417 A225787

Adjacent sequences: A224415 A224416 A224417 * A224419 A224420 A224421

Keyword

nonn

Author

Zhi-Wei Sun, Apr 06 2013

Status

approved