Unordered Factorization

An unordered factorization is a factorization of a number into a product of factors where order is ignored. The following table lists the unordered factorizations of the first few positive integers.

	unordered factorizations
1	1
2	2
3	3
4	, 4
5	5
6	, 6
7	7
8	, , 8
9	, 9
10	, 10

A recurrence product for the number of unordered factorizations is given by Harris and Subbarao (1991).

The numbers of unordered factorizations for , 2, ... are therefore 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, ... (OEIS A001055). The maximum numbers of parts in the unordered (or ordered) factorizations of  for , 2, ... are 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, ... (OEIS A086436).

The following gives a table of unordered factorizations with distinct parts for  between 1 and 10.

	distinct unordered factorizations
1	1
2	2
3	3
4	4
5	5
6	, 6
7	7
8	, 8
9	9
10	, 10

The numbers of unordered factorizations with distinct parts for , 2, ... are given by 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, ... (OEIS A045778). The maximum number of parts in a distinct unordered (or ordered) factorizations of , 2, ... are 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, ... (OEIS A086435).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 844, 1972.

Beckwith, D. "Problem 10669." Amer. Math. Monthly 105, 559, 1998.

Guy, R. K. and Nowakowski, R. J. "Monthly Unsolved Problems, 1969-1995." Amer. Math. Monthly 102, 921-926, 1995.

Harris, V. C. and Subbarao, M. V. "On Product Partitions of Integers." Canad. Math. Bull 34, 474-479, 1991.

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

Sloane, N. J. A. Sequences A001055/M0095, A045778, A086435, and A086436 in "The On-Line Encyclopedia of Integer Sequences."