Numbers whose Divisor Sum is Square

Theorem

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

\(\ds \map {\sigma_1} 3\)

\(=\)

\(\ds 4\)

$\sigma_1$ of $3$

\(\ds \map {\sigma_1} {22}\)

\(=\)

\(\ds 36\)

$\sigma_1$ of $22$

\(\ds \map {\sigma_1} {66}\)

\(=\)

\(\ds 144\)

$\sigma_1$ of $66$

\(\ds \map {\sigma_1} {70}\)

\(=\)

\(\ds 144\)

$\sigma_1$ of $70$

\(\ds \map {\sigma_1} {81}\)

\(=\)

\(\ds 121\)

$\sigma_1$ of $81$

\(\ds \map {\sigma_1} {94}\)

\(=\)

\(\ds 144\)

$\sigma_1$ of $94$

\(\ds \map {\sigma_1} {115}\)

\(=\)

\(\ds 144\)

$\sigma_1$ of $115$

\(\ds \map {\sigma_1} {119}\)

\(=\)

\(\ds 144\)

$\sigma_1$ of $119$

\(\ds \map {\sigma_1} {170}\)

\(=\)

\(\ds 324\)

$\sigma_1$ of $170$

This sequence is A006532 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Examples

$\sigma_1$ of $3$ is Square

$\map {\sigma_1} 3 = 4 = 2^2$

$\sigma_1$ of $22$ is Square

$\map {\sigma_1} {22} = 36 = 6^2$

$\sigma_1$ of $66$ is Square

$\map {\sigma_1} {66} = 144 = 12^2$

$\sigma_1$ of $70$ is Square

$\map {\sigma_1} {70} = 144 = 12^2$

$\sigma_1$ of $81$ is Square

$\map {\sigma_1} {81} = 121 = 11^2$

$\sigma_1$ of $94$ is Square

$\map {\sigma_1} {94} = 144 = 12^2$

$\sigma_1$ of $115$ is Square

$\map {\sigma_1} {115} = 144 = 12^2$

$\sigma_1$ of $119$ is Square

$\map {\sigma_1} {119} = 144 = 12^2$

$\sigma_1$ of $400$ is Square

$\map {\sigma_1} {400} = 961 = 31^2$

Also see

• Square Numbers which are Divisor Sum values

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $66$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $66$