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A350429
Prime numbers p for which there exists at least one integer k < p such that p divides the k-th Bell number.
1
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
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
Sequence in context: A216772 A216734 A114275 * A156107 A267289 A084932
KEYWORD
nonn
AUTHOR
Luis H. Gallardo, Dec 30 2021
STATUS
approved