A286663 a(n) is the least value of k such that for the n-th prime p, p^2 divides Bell(p+k)-Bell(k+1)-Bell(k).

0, 2, 0, 4, 0, 4, 6, 37, 17, 21, 75, 27, 3, 20, 96, 21, 13, 90, 37, 26, 22, 20, 204, 12, 148, 23, 46, 24, 0, 71, 22, 3, 36, 41, 4, 101, 228, 31, 155, 304, 309, 392, 146, 85, 222, 346, 134, 277, 43, 7, 67, 484, 230, 152, 10, 135, 40, 256, 28, 97, 129, 90, 458

Offset

1,2

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.

References

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..100

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

Example

The 7th prime is p(7) = 17, and the least k such that Bell(k+17)-Bell(k)-Bell(k+1) is divisible by 17^2 is k = 6: Bell(23)-Bell(6)-Bell(7) = 44152005855083266 = 17^2*152775106764994, thus a(7) = 6.

Mathematica

a={}; np = 100; p = Prime[Range[np]]; For[i = 0, i < np, i++; p1 = p[[i]];
n = 0; While[!Divisible[BellB[p1 + n] - BellB[n] - BellB[n + 1], p1^2], n++]; a=AppendTo[a, n]]; a

Prog

(PARI) bell(n) = polcoeff(sum(k=0, n, prod(i=1, k, x/(1-i*x)), x^n * O(x)), n);
a(n) = {my(k = 0, p = prime(n)); while ((bell(p+k)-bell(k+1)-bell(k)) % p^2, k++); k; } \\ Michel Marcus, May 20 2017

Crossrefs

Cf. A000110, A283300, A286664.

Sequence in context: A211008 A365344 A366543 * A114402 A035647 A357487

Adjacent sequences: A286660 A286661 A286662 * A286664 A286665 A286666

Keyword

nonn

Author

Amiram Eldar, May 12 2017

Status

approved