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

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