Ordered Factorization

An ordered factorization is a factorization (not necessarily into prime factors) in which a×b is considered distinct from b×a. The following table lists the ordered factorizations for the integers 1 through 10.

n#ordered factorizations
422·2, 4
632·3, 3·2, 6
842·2·2, 2·4, 4·2, 8
923·3, 9
1032·5, 5·2, 10

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


where zeta(s) is the Riemann zeta function.

A recurrence equation for H(n) is given by


where the sum is over the divisors of n and H(1)=1 (Hille 1936, Knopfmacher and Mays 2006). Another recurrence also due to Hille (1936) for n>1 is given by


where H(1)=1/2 and


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

MacMahon (1893) derived the beautiful double sum formula

 H(n)=sum_(j=1)^qsum_(i=0)^(j-1)(-1)^i(j; i)product_(k=1)^r(alpha_k-j-i-1; alpha_k),



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


Chor et al. (2000) showed that

H(n)=2^(alpha_1+alpha_2-1)sum_(k=0)^(alpha_2)(alpha_1; k)(alpha_2; k)2^(-k)

where _2F_1(a,b;c;z) is a hypergeometric function.

The number of ordered factorizations of n is equal to the number of perfect partitions of n-1 (Goulden and Jackson 1983, p. 94).

See also

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

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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.

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Ordered Factorization

Cite this as:

Weisstein, Eric W. "Ordered Factorization." From MathWorld--A Wolfram Web Resource.

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