# Square Numbers whose Sigma is Square

Jump to navigation
Jump to search

## Contents

## 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.