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
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.