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%I #17 Jan 29 2022 10:28:14
%S 5,7,13,19,23,29,37,47,53,61,67,71,73,89,101,107,131,137,139,157,163,
%T 167,173,179,181,191,193,211,223,239,241,251,271,277,281,283,293,307,
%U 311,313,331,349,353,367,401,419,431,433,439,443,449,467,491,499,509,541
%N Prime numbers p for which there exists at least one integer k < p such that p divides the k-th Bell number.
%C Igor Shparlinski proved in 1991 that k < (1/2)*binomial(2*p,p) (see A290059).
%H I. E. Shparlinskiy, <a href="https://doi.org/10.1016/S0195-6698(13)80010-8">On the Distribution of Values of Recurring Sequences and the Bell Numbers in Finite Fields</a>, European Journal of Combinatorics, Vol. 12, No. 1 (1991), pp. 81-87.
%e a(1)=5 since modulo 5 we have B(0)=1, B(1)=1, B(2)=2, and B(3)=0.
%t 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 *)
%Y Cf. A000110, A290059.
%K nonn
%O 1,1
%A _Luis H. Gallardo_, Dec 30 2021
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