Number of ways an integer $n$ can be split into a product of $k$ distinct integers given a prime factorization
I'll show how to solve the case $k=2$ and without the constraint that all factors have to be distinct, and let you generalize and take care of the constraint. Each decomposition corresponds to a choice of non-negative integers $b_1,c_1,b_2,c_2,\ldots,b_i,c_i$ such that $a_j = b_j + c_j$. The number of such decompositions is thus $(a_1+1)\cdots(a_i+1)$.

There might be a slight problem with that, depending on what you mean by decomposition. Consider for example $n=2$. According to the count above, we get the two decompositions $1 \cdot 2$ and $2 \cdot 1$. Perhaps we want to count them as the same decomposition. In this case, we need to roughly divide the count above by 2. Why only roughly? Since if $n = m^2$, then the decomposition $n = m \cdot m$ only appears once. So if $n$ is a square, we need to divide the count above by 2, and add 1/2 (why?).

If we want to count the number of decompositions in which both factors are distinct, we need to check whether $n$ is a square, and then adjust our counts accordingly. Details left to the reader.

For general $k$, if you are OK with ordered decompositions (and ignoring the constraint that all factors be distinct), then all you need do is figure out how many non-negative integer solutions exist for the equation $a = b_1 + \cdots + b_k$. If you care about unordered decompositions (so $1\cdot 2$ and $2\cdot 1$ are considered the same) or require all factors to be distinct, you have more work to do. Inclusion-exclusion type formulas could be helpful.