A224417 - OEIS

%I #9 Apr 06 2013 10:55:39

%S 2,3,2,11,3,2,193,113,2,29,71,167,19,3,7,13,199,5,101,59,13,41,3,359,

%T 7,11,2,31,197,139,3,59,2,139,83,37,23,193,587,199,67,47,401,41,571,

%U 73,1063,229,1163,47,53,239,347,223,577,499,271,269,11,179

%N Least prime p such that sum_{k=0}^n B_k*x^{n-k} is irreducible modulo p, where B_k refers to the Bell number A000110(k).

%C Conjecture: a(n) < 4n^2-1 for all n>0.

%H Zhi-Wei Sun, <a href="/A224417/b224417.txt">Table of n, a(n) for n = 1..300</a>

%e a(5) = 3 since the polynomial sum_{k=0}^5 B_5*x^{5-k} = x^5+x^4+2*x^3+5*x^2+15*x+52 is irreducible modulo 3 but reducible modulo 2.

%e Note also that a(7) = 193 < 4*7^2-1 = 195.

%t A[n_,x_]:=A[n,x]=Sum[BellB[k]*x^(n-k),{k,0,n}]

%t Do[Do[If[IrreduciblePolynomialQ[A[n,x],Modulus->Prime[k]]==True,Print[n," ",Prime[k]];Goto[aa]],{k,1,PrimePi[4n^2-2]}];

%t Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]

%Y Cf. A000040, A000110, A220072, A223934, A224210, A224416, A218465, A217785, A217788, A224197.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Apr 06 2013