Square Numbers whose Sigma is Square

Theorem

The sequence of square numbers whose $\sigma$ value is square starts as follows:

 $\displaystyle \sigma \left({1^2}\right)$ $=$ $\displaystyle 1^2$ $\displaystyle \sigma \left({9^2}\right)$ $=$ $\displaystyle 11^2$ $\displaystyle \sigma \left({20^2}\right)$ $=$ $\displaystyle 31^2$ $\displaystyle \sigma \left({180^2}\right)$ $=$ $\displaystyle 341^2$

Examples

$\sigma$ of $9^2$ is Square

$\sigma \left({9^2}\right) = 11^2$

$\sigma$ of $20^2$ is Square

$\sigma \left({20^2}\right) = 31^2$

$\sigma$ of $180^2$ is Square

$\map \sigma {180^2} = 341^2$

Historical Note

It is reported in 1997: David Wells: Curious and Interesting Numbers (2nd ed.) that this sequence appears in the Journal of Recreational Mathematics, volume $27$, on page $227$.

This is difficult to corroborate, as the author of this page has not been able to find this online.