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A101548
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Number of k such that prime(n) divides the left factorial !k = sum_{i=0..k-1} i!.
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1
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0, 1, 1, 1, 0, 1, 3, 1, 0, 2, 1, 2, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 2, 2, 1, 0, 3, 0, 3, 0, 1, 1, 0, 1, 1, 2, 0, 0, 0, 0, 2, 2, 3, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 4, 1, 2, 0, 1, 3, 0, 1, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,7
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COMMENTS
| Note that 2 divides every left factorial !k for k>1. A result of Barsky and Benzaghou shows that there is no odd prime p such that p divides !p. Hence if an odd prime p divides !k then we must have k < p.
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REFERENCES
| D. Barsky and B. Benzaghou, Nombres de Bell et somme de factorielles, Journal de Theorie des Nombres de Bordeaux, 16:1-17, 2004.
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LINKS
| Bernd C. Kellner, Some remarks on Kurepa's left factorial
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EXAMPLE
| a(8) = 3 because 19 divides !7, !12 and !16.
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MATHEMATICA
| nn=1000; s=0; t=Table[s=s+n!, {n, 0, nn}]; Table[p=Prime[i]; Length[Position[t, _?(0==Mod[ #, p]&)]], {i, 2, PrimePi[nn]}]
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CROSSREFS
| Cf. A003422 (left factorials), A049042 (primes dividing some left factorial), A049043 (primes not dividing any left factorial).
Sequence in context: A051722 A166408 A128618 * A117430 A143676 A002726
Adjacent sequences: A101545 A101546 A101547 * A101549 A101550 A101551
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Dec 06 2004
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