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%I #8 Oct 28 2015 07:03:30
%S 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,
%T 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,
%U 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
%N Number of k such that prime(n) divides the left factorial !k = sum_{i=0..k-1} i!.
%C 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.
%H D. Barsky and B. Benzaghou, <a href="http://dx.doi.org/10.5802/jtnb.432">Nombres de Bell et somme de factorielles</a>, Journal de Théorie des Nombres de Bordeaux, 16:1-17, 2004.
%H Bernd C. Kellner, <a href="http://www.arXiv.org/abs/math.NT/0410477">Some remarks on Kurepa's left factorial</a>, arXiv:math/0410477 [math.NT], 2004.
%e a(8) = 3 because 19 divides !7, !12 and !16.
%t 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]}]
%Y Cf. A003422 (left factorials), A049042 (primes dividing some left factorial), A049043 (primes not dividing any left factorial).
%K nonn
%O 2,7
%A _T. D. Noe_, Dec 06 2004
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