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).
