Left Factorial

The term "left factorial" is sometimes used to refer to the subfactorial , the first few values for , 2, ... are 1, 3, 9, 33, 153, 873, 5913, ... (OEIS A007489).

Unfortunately, the same term and notation are also applied to the factorial sum

			(1)
			(2)
			(3)
			(4)

where  is a gamma function,  is the exponential integral, and  is the En-function.

For , 1, ..., the first few values are given by 0, 1, 2, 4, 10, 34, 154, 874, ... (OEIS A003422). The left factorial is always even for .

 is prime for , 4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354, 721, 1796, 3733, 4769, 9316, 12221, ... (OEIS A100614), the last of which was found by E. W. Weisstein (Oct. 19, 2006).

REFERENCES:

Guy, R. K. Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004.

Kurepa, D. "On the Left Factorial Function ." Math. Balkanica 1, 147-153, 1971.

Kurepa, D. "Left Factorial Function in Complex Domain." Math. Balkanica 3, 297-307, 1973.

Sloane, N. J. A. Sequences A003422/M1237, A007489/M2818, and A100614 in "The On-Line Encyclopedia of Integer Sequences."