Is there any general solution of the equation $x+y+z \mid xyz$ in positive integers where $x,y,z$ are pairwise relatively prime? If $n$ is a certain positive integer, is there any general solution to $x+y+z \mid nxyz$ where they are pairwise relatively prime?
Edit:
For a given $x,y$ we can use $z = xy-x-y$ to give a family of infinite solutions. In fact, if $z=k-x-y$, then: $$k \mid x\cdot y\cdot (k-x-y) \implies k \mid xy(x+y)$$
Thus, the general solution is $(x,y,z) = (x,y,\frac{xy(x+y)}{d}-x-y))$ where $d$ is a divisor of $xy(x+y)$.
Similarly, for a given $n$, we have $(x,y,z) = (x,y,\frac{nxy(x+y)}{d}-x-y))$ where $d$ is a divisor of $nxy(x+y)$.
For the pairwise relatively prime part, is there any addition possible?