Factorial

The factorial  is defined for a positive integer  as

(1)

So, for example, . An older notation for the factorial was written  (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996).

The special case  is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).

The factorial is implemented in the Wolfram Language as Factorial[n] or n!.

The triangular number  can be regarded as the additive analog of the factorial . Another relationship between factorials and triangular numbers is given by the identity

(2)

(K. MacMillan, pers. comm., Jan. 21, 2008).

The factorial  gives the number of ways in which  objects can be permuted. For example, , since the six possible permutations of  are , , , , , . The first few factorials for , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (OEIS A000142).

The numbers of digits in  for , 1, ... are 1, 7, 158, 2568, 35660, 456574, 5565709, 65657060, ... (OEIS A061010).

Generalizations of the factorial such as the double factorial  and multifactorial  can be defined. Note, however, that these are not equal to nested factorials , , etc.

The first few values of  for , 2, ... are 1, 2, 720, 620448401733239439360000, ... (Eureka 1974; OEIS A000197). The numbers of digits in  are 1, 1, 3, 24, 199, 1747, ... (OEIS A063979).

As  grows large, factorials begin acquiring tails of trailing zeros. To calculate the number  of trailing zeros for , use

(3)

where

(4)

and  is the floor function (Gardner 1978, p. 63; Ogilvy and Anderson 1988, pp. 112-114). For , 2, ..., the number of trailing zeros are 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, ... (OEIS A027868). This is a special application of the general result first discovered by Legendre in 1808 that the largest power of a prime  dividing  is

(5)

(Landau 1974, pp. 75-76; Honsberger 1976; Hardy and Wright 1979, pp. 342; Ribenboim 1989; Ingham 1990, p. 20; Graham et al. 1994; Vardi 1991; Hardy 1999, pp. 18 and 21; Havil 2003, p. 165; Boros and Moll 2004, p. 5). This can be implemented in the Wolfram Language as

  HighestPower[p_?PrimeQ, n_] :=
    Sum[Floor[n/p^k], {k, Floor[Log[p,n]]}]

Stated another way, the exact power of a prime  which divides  is

(6)

where  is the digit sum of  in base  (Boros and Moll 2004, p. 6). This can be implemented in the Wolfram Language as

  HighestPower2[p_Integer?PrimeQ, n_] :=
    (n - Total[IntegerDigits[n, p]])/(p - 1)

Therefore, as shown by Legendre,

(7)

(Havil 2003, p. 165).

Let  be the last nonzero digit in , then the first few values are 2, 6, 4, 2, 2, 4, 2, 8, 8, 8, 6, 8, ... (OEIS A008904). This sequence was studied by Kakutani (1967), who showed that this sequence is "5-automatic," meaning roughly that there exists a finite automaton which, when given the digits of  in base-5, will wind up in a state for which an output mapping specifies . The exact distribution of digits follows from this result.

Min   Max       Re       Im

By noting that

(8)

where  is the gamma function for integers , the definition can be generalized to complex values

(9)

This defines  for all complex values of , except when  is a negative integer, in which case  is equal to complex infinity.

While Gauss (G1) introduced the notation

(10)

this notation was subsequently abandoned after Legendre introduced the gamma-notation (Edwards 2001, p. 8).

Using the identities for gamma functions, the values of  (half integral values) can be written explicitly

			(11)
			(12)
			(13)
			(14)

where  is a double factorial.

For integers  and  with ,

(15)

The logarithm of  is frequently encountered

			(16)
			(17)
			(18)
			(19)
			(20)

where  is the Euler-Mascheroni constant,  is the Riemann zeta function, and  is the polygamma function.

It is also given by the limit

			(21)
			(22)
			(23)
			(24)

where  is the Pochhammer symbol.

where  is the Euler-Mascheroni constant,  is the Riemann zeta function, and  is the polygamma function. The factorial can be expanded in a series

(25)

(OEIS A001163 and A001164). Stirling's series gives the series expansion for ,

			(26)
			(27)

(OEIS A046968 and A046969), where  is a Bernoulli number.

In general, the power-product sequences (Mudge 1997) are given by . The first few terms of  are 2, 5, 37, 577, 14401, 518401, ... (OEIS A020549), and  is prime for , 2, 3, 4, 5, 9, 10, 11, 13, 24, 65, 76, ... (OEISA046029). The first few terms of  are 0, 3, 35, 575, 14399, 518399, ... (OEIS A046032), but  is prime for only  since  for . The first few terms of  are 0, 7, 215, 13823, 1727999, ... (OEIS A046033), and the first few terms of  are 2, 9, 217, 13825, 1728001, ... (OEIS A019514).

The first few numbers  such that the sum of the factorials of their digits is equal to the prime counting function  are 6500, 6501, 6510, 6511, 6521, 12066, 50372, ... (OEIS A049529). This sequence is finite, with the largest term being .

Numbers  such that

(28)

are called Wilson primes.

Brown numbers are pairs  of integers satisfying the condition of Brocard's problem, i.e., such that

(29)

Only three such pairs are known: (5, 4), (11, 5), (71, 7). Erdős conjectured that these are the only three such pairs (Guy 1994, p. 193).

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