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A224480 Least prime q such that x^n + sum_{k=1}^n p_k*x^{n-k} is irreducible modulo q, where p_k denotes the k-th prime.

%I A224480 #9 Apr 08 2013 02:48:39
%S A224480 2,11,2,2,2,2,2,53,13,3,5,2,2,2,2,421,29,19,7,2,29,37,2,743,41,23,13,
%T A224480 47,5,2,269,139,211,31,73,307,2,2,5,89,23,839,181,379,173,89,2,353,
%U A224480 101,307,3,29,389,2,863,71,503,619,193,2
%N A224480 Least prime q such that x^n + sum_{k=1}^n p_k*x^{n-k} is irreducible modulo q, where p_k denotes the k-th prime.
%C A224480 Conjecture: a(n) <= (n+4)*(n+5)+1 for all n>0.
%H A224480 Zhi-Wei Sun, <a href="/A224480/b224480.txt">Table of n, a(n) for n = 1..500</a>
%e A224480 a(10) = 3 since P(x) = x^{10} + 2*x^9 + 3*x^8 + 5*x^7 + 7*x^6
%e A224480 + 11*x^5 + 13*x^4 + 17*x^3 + 19*x^2 + 23*x + 29 is irreducible modulo 3, but reducible modulo 2, for,
%e A224480     P(x)==(x+1)^2*(x^3+x+1)*(x^5+x^3+1) (mod 2).
%e A224480 Note also that a(16) = 421 = (16+4)*(16+5)+1.
%t A224480 A[n_,x_]:=A[n,x]=Sum[x^n+Prime[k]*x^(n-k),{k,1,n}]
%t A224480 Do[Do[If[IrreduciblePolynomialQ[A[n,x],Modulus->Prime[k]]==True,Print[n," ",Prime[k]];Goto[aa]],{k,1,PrimePi[n^2+9n+21]}];
%t A224480 Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]
%Y A224480 Cf. A000040, A220947, A220072, A223934, A224210, A224416, A224417, A224418, A217788, A224197, A218465, A217785.
%K A224480 nonn
%O A224480 1,1
%A A224480 _Zhi-Wei Sun_, Apr 07 2013