An unordered factorization, also called a multiplicative partition, is a factorization of a positive integer into a product of integer factors greater than 1, with the order of the factors ignored. By convention, 1 has the single factorization consisting of the one part 1. 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 product recurrence for the number of unordered factorizations is given by Harris and Subbarao (1991, p. 477).
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 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
numbers of parts in distinct unordered 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,
... With the convention above, the first maximum is 1. For 
, the maxima agree with OEIS A086435,
which uses an empty factorization and a maximum
of 0 for 
.
For 
,
both maxima also apply to ordered factorizations.
