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

Author

Luis H. Gallardo, Dec 30 2021

Keywords

• nonn

Comments

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

Examples

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

Links

• 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.

Crossrefs

Cf. A000110, A290059.

Programs

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 *)