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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).
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%I #11 Aug 04 2019 20:04:00

%S 1,1,4,96,356540,39903286,1312583081304,356826497344324,

%T 51202108292508282304,10903333036235662560405182340,

%U 8851961858819132893480466080328,10341369256681418109100257759613689061054,20410983764150196478167108200311379711212644128

%N 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).

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

%H Daniel Barsky and Bénali Benzaghou, <a href="http://www.numdam.org/item/JTNB_2004__16_1_1_0/">Nombres de Bell et somme de factorielles</a>, Journal de théorie des nombres de Bordeaux, Vol. 16, No. 1 (2004), pp. 1-17.

%H Anne Gertsch Hamadene, <a href="http://doc.rero.ch/record/4372/files/2_these_GertschHamadeneA.pdf">Congruences pour quelques suites classiques de nombres, sommes de factorielles et calcul ombral</a>, Doctoral dissertation, Université de Neuchâtel, 1999.

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

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

%o (PARI) a(n) = my(p=prime(n)); (a003422(p) - a000110(p-1) + 1)/p \\ _Felix Fröhlich_, Aug 04 2019

%Y Cf. A000110, A003422, A079609.

%K nonn

%O 1,3

%A _Amiram Eldar_, Aug 04 2019