An ordered factorization of a positive integer is an ordered list of integer factors greater than 1 whose product is , with rearrangements counted as distinct when the lists differ. The factors need not be prime. For , the empty product is counted as one ordered factorization. The following table lists the ordered factorizations for the integers 1 through 10.

	#	ordered factorizations
1	1	1
2	1	2
3	1	3
4	2	, 4
5	1	5
6	3	, , 6
7	1	7
8	4	, , , 8
9	2	, 9
10	3	, , 10

The numbers of ordered factorizations of , 2, ... are given by 1, 1, 1, 2, 1, 3, 1, 4, 2, 3, ... (OEIS A074206). This sequence has the Dirichlet generating function

(1)

for , where is the real part of , is the Riemann zeta function, and is the unique root of .

A recurrence equation for is given by

(2)

for , with (Hille 1936, Knopfmacher and Mays 2006). Another recurrence due to Hille (1936) uses and for . It is given by

(3)

for , where is the prime factorization of , the are distinct primes, and the sums run over distinct prime subsets, with in the second sum (Knopfmacher and Mays 2006).

MacMahon (1893) derived the beautiful double sum formula

(4)

where and (Knopfmacher and Mays 2006). For with distinct primes and positive integers , Chor et al. (2000) showed that

			(5)
			(6)

where is a hypergeometric function.

The number of ordered factorizations of is equal to the number of perfect partitions of (Goulden and Jackson 1983, p. 94).