The authors prove that the size of the set of \(k\)-perfect numbers with \(r\) distinct prime factors is bounded above by \((k-1)4^{r^{3}}\), generalizing Paul Pollack’s bound in the case when \(k=2\). The proof follows Pollack’s methods. The new key step involves a reduction to “primitive” \(\alpha\)-perfect numbers.
It is interesting to note that this bound on the number of \(k\)-perfect integers with \(r\) distinct prime factors is significantly smaller than the best known bound on the size of any such number.