A286664 a(n) is the smallest prime p such that p^2 divides Bell(p+n) - Bell(n+1) - Bell(n).

2, 5, 2, 2, 2, 20663, 2, 229, 2, 2, 2, 11, 2, 5, 2, 2, 2, 23, 2, 3, 2, 2, 2, 101, 2, 3, 2, 2, 2

Offset

0,1

Comments

Jacques Touchard proved in 1933 that for the Bell numbers (A000110), Bell(p+k) == Bell(k+1) + Bell(k) (mod p) for all primes p and k >= 0.

a(29) > 242000 and a(89) > 90000, if they exist. The terms from a(30) to a(89) are 2, 163, 2, 2, 2, 7, 2, 19, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 7, 2, 2, 2, 359, 2, 3, 2, 2, 2, 7, 2, 43, 2, 2, 2, 3, 2, 5, 2, 2, 2, 5, 2, 547, 2, 2, 2, 3, 2, 7, 2, 2, 2, 59, 2, 5, 2, 2, 2. - Giovanni Resta, Aug 26 2018

a(29) > 10^7. - Hiroaki Yamanouchi, Sep 01 2018

References

J. Touchard, "Propriétés arithmétiques de certains nombres récurrents", Ann. Soc. Sci. Bruxelles A 53 (1933), pp. 21-31.

Links

Table of n, a(n) for n=0..28.

Eric Weisstein's World of Mathematics, Touchard's Congruence

Example

The smallest prime p such that Bell(p+1) == Bell(2)+Bell(1)(mod p^2) is 5, since Bell(6) - Bell(2) - Bell(1) = 203 - 2 - 1 = 200 = 5^2 * 8, thus a(1) = 5.

Mathematica

a = {}; n = 0; While[n < 101, p = 2; While[!Divisible[BellB[p + n] - BellB[n] - BellB[n + 1], p^2], p = NextPrime@p]; a = AppendTo[a, p]; n++]; a

Crossrefs

Cf. A000110, A283300, A286663.

Sequence in context: A226029 A152765 A327867 * A306541 A242242 A171937

Adjacent sequences: A286661 A286662 A286663 * A286665 A286666 A286667

Keyword

nonn,more

Author

Amiram Eldar, May 12 2017

Status

approved