A309483 Quotients (!p - Bell(p-1) + 1)/p where p is the n-th prime, !k is Kurepa's left-factorial function (A003422) and Bell(k) is the k-th Bell number (A000110).

1, 1, 4, 96, 356540, 39903286, 1312583081304, 356826497344324, 51202108292508282304, 10903333036235662560405182340, 8851961858819132893480466080328, 10341369256681418109100257759613689061054, 20410983764150196478167108200311379711212644128

Offset: 1

Author

Amiram Eldar, Aug 04 2019

Keywords

• nonn

Comments

Gertsch Hamadene proved that !p == Bell(p-1) - 1 (mod p) for primes p.

Examples

			The 3rd prime is 5, so a(3) = (!5 - Bell(5-1) + 1)/5 = (34 - 15 + 1)/5 = 4.
		

Links

• Daniel Barsky and Bénali Benzaghou, Nombres de Bell et somme de factorielles, Journal de théorie des nombres de Bordeaux, Vol. 16, No. 1 (2004), pp. 1-17.
• Anne Gertsch Hamadene, Congruences pour quelques suites classiques de nombres, sommes de factorielles et calcul ombral, Doctoral dissertation, Université de Neuchâtel, 1999.

Crossrefs

Cf. A000110, A003422, A079609.

Programs

Mathematica

quot[p_] := (Sum[k!, {k, 0, p - 1}] - BellB[p - 1] + 1)/p; Table[quot[Prime[i]], {i, 1, 13}]

PARI

a(n) = my(p=prime(n)); (a003422(p) - a000110(p-1) + 1)/p \\ Felix Fröhlich, Aug 04 2019