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 A101548 Number of k such that prime(n) divides the left factorial !k = sum_{i=0..k-1} i!. 1
 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; text; internal format)
 OFFSET 2,7 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. LINKS D. Barsky and B. Benzaghou, Nombres de Bell et somme de factorielles, Journal de ThÃ©orie des Nombres de Bordeaux, 16:1-17, 2004. Bernd C. Kellner, Some remarks on Kurepa's left factorial, arXiv:math/0410477 [math.NT], 2004. EXAMPLE a(8) = 3 because 19 divides !7, !12 and !16. 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]}] CROSSREFS Cf. A003422 (left factorials), A049042 (primes dividing some left factorial), A049043 (primes not dividing any left factorial). Sequence in context: A166408 A128618 A284826 * A117430 A143676 A002726 Adjacent sequences:  A101545 A101546 A101547 * A101549 A101550 A101551 KEYWORD nonn AUTHOR T. D. Noe, Dec 06 2004 STATUS approved

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Last modified December 14 04:53 EST 2018. Contains 318090 sequences. (Running on oeis4.)