I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^p$$ such that $m=k{n}^{p-1}$ with $m,n>0$ and $p$ is an odd prime?

Note: $\sigma(\frac{m}{{n}^{p-1}})$ is the sum of divisors function of $\frac{m}{ {n}^{p-1}}$ such that $m=k{n}^{p-1}$.

**Edit 2**: I edited the question as it has the same meaning for the precedent.

Thank you for any help.

rationalsquare. Perhaps this already follows from known results. $\endgroup$ – Noam D. Elkies Jul 29 '15 at 4:37