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A309483
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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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0
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1, 1, 4, 96, 356540, 39903286, 1312583081304, 356826497344324, 51202108292508282304, 10903333036235662560405182340, 8851961858819132893480466080328, 10341369256681418109100257759613689061054, 20410983764150196478167108200311379711212644128
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OFFSET
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1,3
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COMMENTS
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Gertsch Hamadene proved that !p == Bell(p-1) - 1 (mod p) for primes p.
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LINKS
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EXAMPLE
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The 3rd prime is 5, so a(3) = (!5 - Bell(5-1) + 1)/5 = (34 - 15 + 1)/5 = 4.
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MATHEMATICA
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quot[p_] := (Sum[k!, {k, 0, p - 1}] - BellB[p - 1] + 1)/p; Table[quot[Prime[i]], {i, 1, 13}]
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PROG
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(PARI) a(n) = my(p=prime(n)); (a003422(p) - a000110(p-1) + 1)/p \\ Felix Fröhlich, Aug 04 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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