Factorial Sums

The sum-of-factorial powers function is defined by

(1)

For ,

			(2)
			(3)
			(4)

where is the exponential integral, (OEIS A091725), is the En-function, is the real part of , and i is the imaginary number. The first few values are 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, ... (OEIS A007489). cannot be written as a hypergeometric term plus a constant (Petkovšek et al. 1996). The only prime of this form is , since

			(5)
			(6)
			(7)

is always a multiple of 3 for .

In fact, is divisible by 3 for and , 5, 7, ... (since the Cunningham number given by the sum of the first two terms is always divisible by 3--as are all factorial powers in subsequent terms ) and so contains no primes, meaning sequences with even are the only prime contenders.

The sum

(8)

does not appear to have a simple closed form, but its values for , 2, ... are 1, 5, 41, 617, 15017, 533417, 25935017, ... (OEIS A104344). It is prime for indices 2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, 32841, ... (OEIS A100289). Since is divisible by 1248829 for , there can be only a finite number of such primes. (However, the largest such prime is not known, which is not surprising given that has more than 14 million decimal digits.)

is divisible by 13 for and the only prime with is .

The case of is slightly more interesting, but is divisible by 1091 for and checking the terms below that gives the only prime terms as , 34, and 102 (OEIS A289947).

The only prime in is for since is divisible by 13 for .

Similarly, the only primes in are for , 4, 5, 16, and 25 (OEIS A290014). since is divisible by 41 for .

The sequence of smallest (prime) numbers such that is divisible by for is given for , 2, ... by 1248829, 13, 1091, 13, 41, 37, 463, 13, 23, 13, 1667, 37, 23, 13, 41, 13, 139, ... (OEIS A290250).

The related sum with index running from 0 instead of 1 is sometimes denoted (not to be confused with the subfactorial) and known as the left factorial,

(9)

The related sum with alternating terms is known as the alternating factorial,

(10)

The sum

(11)

has a simple form, with the first few values being 1, 5, 23, 119, 719, 5039, ... (OEIS A033312).

Identities satisfied by sums of factorials include

			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)

(OEIS A001113, A068985, A070910, A091681, A073743, A049470, A073742, and A049469; Spanier and Oldham 1987), where is a modified Bessel function of the first kind, is a Bessel function of the first kind, is the hyperbolic cosine, is the cosine, is the hyperbolic sine, and is the sine.

Sums of factorial powers include

			(20)
			(21)
			(22)
			(23)

(OEIS A091682 and A091683) and, in general,

(24)

Schroeppel and Gosper (1972) give the integral representation

(25)

where

			(26)
			(27)
			(28)

There are only four integers equal to the sum of the factorials of their digits. Such numbers are called factorions.

While no factorial greater than 1! is a square number, D. Hoey listed sums of distinct factorials which give square numbers, and J. McCranie gave the one additional sum less than :

			(29)
			(30)
			(31)
			(32)
			(33)
			(34)
			(35)
			(36)
			(37)
			(38)
			(39)
			(40)
			(41)
			(42)

and

(43)

(OEIS A014597).

Sums with powers of an index in the numerator and products of factorials in the denominator can often be done analytically in terms of regularized hypergeometric functions , for example

(44)

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