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# 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).

Distinct Prime Factorization, Factorization, Perfect Partition, Prime Factorization, Unordered Factorization

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## 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.

## Referenced on Wolfram|Alpha

Ordered Factorization

## Cite this as:

Weisstein, Eric W. "Ordered Factorization." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrderedFactorization.html