An ordered factorization of a positive integer 
is an ordered list of integer
factors greater than 1 whose product is 
, with rearrangements counted as distinct when the lists differ.
The factors need not be prime. For 
, the empty product is counted
as one ordered factorization. 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)
|
for 
,
where 
is the real part of 
, 
is the Riemann zeta
function, and 
is the unique root of 
.
A recurrence equation for 
is given by
 |
(2)
|
for 
,
with 
(Hille 1936, Knopfmacher and Mays 2006). Another recurrence due to Hille (1936) uses
and 
for 
.
It is given by
![H^^(n)=2[sum_(p_i)H^^(n/(p_i))-sum_(p_i,p_j)H^^(n/(p_ip_j))+...+(-1)^(r-1)H^^(n/(p_1p_2...p_r))]](/images/equations/OrderedFactorization/NumberedEquation3.svg) |
(3)
|
for 
,
where 
is the prime factorization of 
, the 
are distinct primes, and
the sums run over distinct prime subsets, with 
in the second sum (Knopfmacher and Mays 2006).
MacMahon (1893) derived the beautiful double sum formula
 |
(4)
|
where 
and 
(Knopfmacher and Mays 2006). For 
with distinct primes 
and positive
integers 
,
Chor et al. (2000) showed that
 |  |  |
(5)
|
 |  |  |
(6)
|
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).
