cp's OEIS Frontend

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).

%I A224418 #13 Apr 06 2013 10:56:15
%S A224418 2,3,2,11,2,13,19,19,13,29,73,47,19,43,7,59,13,29,3,13,179,29,173,19,
%T A224418 3,163,23,3,101,71,131,977,5,157,43,13,73,2,89,197,151,151,313,3,13,
%U A224418 31,23,97,173,241,181,109,487,157,17,29,89,109,257,317
%N 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).
%C A224418 Conjecture: a(n) < n^2 for all n > 1.
%H A224418 Zhi-Wei Sun, <a href="/A224418/b224418.txt">Table of n, a(n) for n = 1..400</a>
%e A224418 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.
%t A224418 A[n_,x_]:=A[n,x]=Sum[PartitionsP[k]*x^(n-k),{k,0,n}]
%t A224418 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]]}];
%t A224418 Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]
%Y A224418 Cf. A000040, A000041, A224417, A224416, A220072, A223934, A224210, A217788, A224197.
%K A224418 nonn
%O A224418 1,1
%A A224418 _Zhi-Wei Sun_, Apr 06 2013