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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
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 Théorie 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: A327077 A284826 A307752 * A117430 A354795 A143676
KEYWORD
nonn
AUTHOR
T. D. Noe, Dec 06 2004
STATUS
approved