This question is related to optimizing the amount of computations of a program.
How many sets of $m$ naturals can be formed such that their product equals the number $n$ ?
Here is what I tried
Let $\prod_{i=1}^\ell p_i^{k_i}$ be the prime factorization of $n$. Then we want to distribute the $k_i$ $p_i$s to the $m$ numbers in the set for every $i$. What we can do is making a list of $k_i$ stars and $m-1$ bars and choosing the placement of the $m-1$ bars among this list, this gives me ${k_i+m-1\choose m-1}$ ways of choosing how to dispatch the $k_i$ $p_i$s to those $m$ numbers.
Then I multiply all these for any $i$ and since I don't care about rearrangement of the $m$ numbers. So my answer is
$$\frac{1}{m!} \prod_{i=1}^\ell {k_i+m-1\choose m-1},$$
which unfortunately is not necessarily an integer and doesn't match what I found manually. Does any of you have an idea ?
