Are there any perfect numbers which are also powerful?

Question

Powerful numbers are discussed in this paper by R. A. Mollin and P. G. Walsh.

Wikipedia has more information.

In particular, note that OEIS A001694 does not seem to contain any (even) perfect numbers.

This is indeed the case, if we note that the Mersenne prime $2^p - 1$ has exponent $1$. (Recall that the general form for even perfect numbers is $N = {2^{p - 1}}(2^p - 1)$.)

Now my question is: How about odd perfect numbers?

I guess my main inquiry boils down to - Has anyone worked on considering properties of the set $O \bigcap P,$ where $O$ and $P$ are given as follows

$$O = \{\text{odd perfect numbers}\}$$ $$P = \{\text{powerful numbers}\}$$

Any pointers to existing references in the literature will be appreciated.

Thank you!

Answer 1

It is known that any odd perfect number is of the form $p^rn^2$ where $p$ is a prime, $p$ doesn't divide $n$, and $r\equiv1\pmod4$. The case $r=1$ has not been ruled out, so it has not been proved that every odd perfect is powerful; the cases $r\gt1$ have not been ruled out, so it has not been proved that no odd perfect number is powerful.