Numbers whose Divisor Sum is Square
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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
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): $66$