(Note: The following post is an offshoot of this earlier MSE question: Can a multiperfect number be a perfect square?.)
Let $\sigma(x)$ denote the classical sum of divisors of the positive integer $x$.
A number $m$ satisfying $$\sigma(m)=2m$$ is said to be perfect.
More generally, we call any number $n$ satisfying $$\sigma(n)=kn$$ for $k \in \mathbb{N}$ to be multiperfect (or $k$-perfect).
It is known that multiperfect numbers cannot be squares.
Furthermore, it is also known that perfect numbers cannot be perfect powers.
I found a reference to the last statement in Walter Nissen's Concise, remarkable facts about perfect numbers:
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Perfect Naturals
are not
Perfect Powers ( e.g. , perfect squares , perfect cubes , etc. )
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Here is my question:
Can a multiperfect number be a perfect power?
Update (August 9, 2020 - 12:04 PM Manila time) I have posted a closely related question in MO here.
I checked all the multiperfect numbers from this site, which include all the multiperfect numbers in the b-file of A007691, as well as additional ones. The largest number that was in the list was $\approx 10^{34850339}$. None of those values were perfect powers.
What I did find for all multiperfect numbers is that there was at least one prime factor with an exponent of exactly $1$ in the prime factorization. If this could be proved for all multiperfect numbers, then the conjecture that there is no number that is both multiperfect and a perfect power could be proven.