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

1,1

Comments

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

Links

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A241055 A224418 A220947 * A225787 A012960 A013117

Adjacent sequences: A224414 A224415 A224416 * A224418 A224419 A224420

Keyword

nonn

Author

Zhi-Wei Sun, Apr 06 2013

Status

approved