Square Numbers whose Sigma is Square

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Theorem

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

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

This sequence is A008847 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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

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


Also see


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.


Sources