$\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$
This is one of the popular equation to find out the number of solutions. From Google, here I found that for equation $\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$, number of solutions are
$\psi(n)=\text{number of divisors of } n$.
$\frac{\psi(n^2)+1}{2}$
but when i turned here, it says that for equation $\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$, number of solutions are $\frac{\psi(n!^2)-1}{2}$
However in the first link they explained the equation for $n=4$ and their formula correctly suits on that. But I want to confirm the correct answer.
If there exist any better way to get the number of positive integral solutions for the equation $\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$, then please suggest me.
Would prime factorization help here? I have an algorithm to find out prime factors of any number. But not have any exact idea about implementing it over here.
My objective is to find out total number of positive integral solutions of the equation $\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$.
Thank you.