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Date: 21-10-2020
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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.
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