Is There some one who can show me if there are infinitely many $k$ for which
$$\frac{\sigma(k)}{k}$$ is a rational square where $\sigma(k)$ and $k$ both are square ?
Note :$\sigma(k)$ is sum divisor function of $k$
Thank you for any help
2026-03-29 16:35:45.1774802145
Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}$ is a rational square where $ \sigma(k) $ and $k$ both are square?
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This is completely unknown.
Your question boils down to whether $\sigma(n^2)$ is square infinitely often, and we have no idea. The OEIS has a list of known cases, but it is not known to be infinite. I would say there is not even a hint of an idea as to how to proceed towards proving there are or are not infinitely many, and both directions are completely unknown.