A101548 Number of k such that prime(n) divides the left factorial !k = sum_{i=0..k-1} i!.

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

Author

T. D. Noe, Dec 06 2004

Keywords

• nonn

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.

Examples

			a(8) = 3 because 19 divides !7, !12 and !16.
		

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.

Crossrefs

Cf. A003422 (left factorials), A049042 (primes dividing some left factorial), A049043 (primes not dividing any left factorial).

Programs

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]}]