Consider the prime factorization of a natural number $N$.

$N=p_1^{e_1}\times p_2^{e_2}\times p_3^{e_3}\times\cdots{ } \cdots{ } \cdots{ }\times p_r^{e_r}$

I need to know-

How many ways are there to express $N$ as a product of $k\geq2$ positive integers, with ignoring order of the factors?

That means,

How many distinct choices do we have for the ordered $k$-tuple $(a_1,a_2, \cdots{ } \cdots{ } \cdots{ },a_k),$ with each $a_i$ is a positive integer such that

$N=a_1a_2\cdots{ } \cdots{ } \cdots{ }a_k$ and $a_1\leq a_2\leq \cdots{ } \cdots{ } \cdots{ }\leq a_k$.

For $k=2$, the number of choices is: $\frac{d(N)+1}{2}$ if $N$ is a perfect square, or $\frac{d(N)}{2}$ if $N$ is not a perfect square. Here $d(N)$ is the number of factors of $N$.

I'm totally in dark about how to proceed.

Any help will be greatly appreciated.

I prefer a solution that uses the prime factorization of $N$. But any other approach is also much welcome.


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