Ordered Factorization

An ordered factorization is a factorization (not necessarily into prime factors) in which  is considered distinct from . 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)

where  is the Riemann zeta function.

A recurrence equation for  is given by

(2)

where the sum is over the divisors of  and  (Hille 1936, Knopfmacher and Mays 2006). Another recurrence also due to Hille (1936) for  is given by

(3)

where  and

(4)

is the prime factorization of  (Knopfmacher and Mays 1996).

MacMahon (1893) derived the beautiful double sum formula

(5)

where

(6)

(Knopfmacher and Mays 1996). In the case that  is a product of two prime powers,

(7)

Chor et al. (2000) showed that

			(8)
			(9)

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).

REFERENCES:

Chor, B.; Lemke, P.; and Mador, Z. "On the Number of Ordered Factorizations of Natural Numbers." Disc. Math. 214, 123-133, 2000.

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 126, 1974.

Goulden, I. P. and Jackson, D. M. Problem 2.5.12 in Combinatorial Enumeration. New York: Wiley, p. 94, 1983.

Hille, E. "A Problem in 'Factorisatio Numerorum.' " Acta Arith. 2, 134-144, 1936.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 141, 1985.

Knopfmacher, A. and Mays, M. "Ordered and Unordered Factorizations of Integers." Mathematica J. 10, 72-89, 2006.

MacMahon, P. A. "Memoir on the Theory of the Compositions of Numbers." Philos. Trans. Roy. Soc. London (A) 184, 835-901, 1893.

Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 124, 1980.

Sloane, N. J. A. Sequence A074206 in "The On-Line Encyclopedia of Integer Sequences."

Warlimont, R. "Factorisatio Numerorum with Constraints." J. Number Th. 45, 186-199, 1993.