A224417 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).

2, 3, 2, 11, 3, 2, 193, 113, 2, 29, 71, 167, 19, 3, 7, 13, 199, 5, 101, 59, 13, 41, 3, 359, 7, 11, 2, 31, 197, 139, 3, 59, 2, 139, 83, 37, 23, 193, 587, 199, 67, 47, 401, 41, 571, 73, 1063, 229, 1163, 47, 53, 239, 347, 223, 577, 499, 271, 269, 11, 179

Offset: 1

Author

Zhi-Wei Sun, Apr 06 2013

Keywords

• nonn

Comments

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

Examples

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

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Crossrefs

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

Programs

Mathematica

A[n_,x_]:=A[n,x]=Sum[BellB[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[4n^2-2]}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]