# Can a multiperfect number be a perfect power?

(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.

• Just to mention it : $1$ is not considered to be a multi-perfect number. Currently, I am checking the cubes upto some limit, no idea yet for a proof. – Peter Aug 7 '20 at 22:07
• No cube upto $10^{21}$ is multi-perfect. No $5$ th power upto $10^{30}$ is multi-perfect. – Peter Aug 7 '20 at 22:13
• I checked all the multi-perfect numbers in the b-file of A007691, and none of them were perfect powers. This is all the way up to $n \approx 1.8 \cdot 10^{303}$ – Varun Vejalla Aug 7 '20 at 22:35
• @VarunVejalla: Please post your last comment as an answer, so that I can upvote it. Thank you. – Arnie Bebita-Dris Aug 8 '20 at 0:24

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.