Divisors of $q^kp^r$

Divisors of $q^kp^r$

This is a generalization of my previous problem. Let $p$ and $q$ be prime
numbers. What is the necessary and sufficient condition (in terms of $p,q$
and $k,r$) such that we can partition the divisors of $q^kp^r$ into two
sets with equal sum ?
For $p \not = q=2$ it is proved that $r$ must be odd and $q^{k+1} > p$.
See Here