# Square Numbers whose Divisor Sum is Square

## Theorem

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

 $\ds \map {\sigma_1} {1^2}$ $=$ $\ds 1^2$ $\ds \map {\sigma_1} {9^2}$ $=$ $\ds 11^2$ $\ds \map {\sigma_1} {20^2}$ $=$ $\ds 31^2$ $\ds \map {\sigma_1} {180^2}$ $=$ $\ds 341^2$ $\ds \map {\sigma_1} {1306^2}$ $=$ $\ds 1729^2$

## 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

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

## 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.