Square Numbers whose Divisor Sum is Square

Theorem

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

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

\(=\)

\(\ds 1^2\)

\(\ds \map {\sigma_1} {9^2}\)

\(=\)

\(\ds 11^2\)

\(\ds \map {\sigma_1} {20^2}\)

\(=\)

\(\ds 31^2\)

\(\ds \map {\sigma_1} {180^2}\)

\(=\)

\(\ds 341^2\)

\(\ds \map {\sigma_1} {1306^2}\)

\(=\)

\(\ds 1729^2\)

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

• Numbers whose Divisor Sum is Square
• Square Numbers which are Divisor Sum values

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.