A350429 Prime numbers p for which there exists at least one integer k < p such that p divides the k-th Bell number.

5, 7, 13, 19, 23, 29, 37, 47, 53, 61, 67, 71, 73, 89, 101, 107, 131, 137, 139, 157, 163, 167, 173, 179, 181, 191, 193, 211, 223, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 313, 331, 349, 353, 367, 401, 419, 431, 433, 439, 443, 449, 467, 491, 499, 509, 541

Offset

1,1

Comments

Igor Shparlinski proved in 1991 that k < (1/2)*binomial(2*p,p) (see A290059).

Links

Table of n, a(n) for n=1..56.

I. E. Shparlinskiy, On the Distribution of Values of Recurring Sequences and the Bell Numbers in Finite Fields, European Journal of Combinatorics, Vol. 12, No. 1 (1991), pp. 81-87.

Example

a(1)=5 since modulo 5 we have B(0)=1, B(1)=1, B(2)=2, and B(3)=0.

Mathematica

q[p_] := Module[{k = 1}, While[k < p && ! Divisible[BellB[k], p], k++]; k < p]; Select[Range[500], PrimeQ[#] && q[#] &] (* Amiram Eldar, Dec 30 2021 *)

Crossrefs

Cf. A000110, A290059.

Sequence in context: A216772 A216734 A114275 * A156107 A267289 A084932

Adjacent sequences: A350426 A350427 A350428 * A350430 A350431 A350432

Keyword

nonn

Author

Luis H. Gallardo, Dec 30 2021

Status

approved