# Square Numbers whose Divisor Sum is Square

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## Theorem

The sequence of square numbers whose divisor sum is square starts as follows:

\(\ds \map {\sigma_1} {1^2}\) | \(=\) | \(\ds 1^2\) | ||||||||||||

\(\ds \map {\sigma_1} {9^2}\) | \(=\) | \(\ds 11^2\) | ||||||||||||

\(\ds \map {\sigma_1} {20^2}\) | \(=\) | \(\ds 31^2\) | ||||||||||||

\(\ds \map {\sigma_1} {180^2}\) | \(=\) | \(\ds 341^2\) | ||||||||||||

\(\ds \map {\sigma_1} {1306^2}\) | \(=\) | \(\ds 1729^2\) |

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

## Examples

### $\sigma_1$ of $9^2$ is Square

- $\map {\sigma_1} {9^2} = 11^2$

### $\sigma_1$ of $20^2$ is Square

- $\map {\sigma_1} {20^2} = 31^2$

### $\sigma_1$ of $180^2$ is Square

- $\map {\sigma_1} {180^2} = 341^2$

### $\sigma_1$ of $1306^2$ is Square

- $\map {\sigma_1} {1306^2} = 1729^2$

## 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.