Square Numbers which are Divisor Sum values
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Theorem
The sequence of square numbers which are the divisor sum value of a (strictly) positive integer begins:
- $1, 4, 36, 121, 144, 256, 324, 400, 576, 784, 900, 961, \ldots$
This sequence is A038688 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds 1\) | \(=\) | \(\ds \map {\sigma_1} 1\) | $\sigma_1$ of $1$ | |||||||||||
\(\ds 4\) | \(=\) | \(\ds \map {\sigma_1} 3\) | $\sigma_1$ of $3$ | |||||||||||
\(\ds 36\) | \(=\) | \(\ds \map {\sigma_1} {22}\) | $\sigma_1$ of $22$ | |||||||||||
\(\ds 121\) | \(=\) | \(\ds \map {\sigma_1} {81}\) | $\sigma_1$ of $81$ | |||||||||||
\(\ds 144\) | \(=\) | \(\ds \map {\sigma_1} {66}\) | $\sigma_1$ of $66$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {70}\) | $\sigma_1$ of $70$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {94}\) | $\sigma_1$ of $94$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {115}\) | $\sigma_1$ of $115$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {119}\) | $\sigma_1$ of $119$ | |||||||||||
\(\ds 256\) | \(=\) | \(\ds \map {\sigma_1} {217}\) | $\sigma_1$ of $217$ | |||||||||||
\(\ds 324\) | \(=\) | \(\ds \map {\sigma_1} {170}\) | $\sigma_1$ of $170$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {214}\) | $\sigma_1$ of $214$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {265}\) | $\sigma_1$ of $265$ | |||||||||||
\(\ds 400\) | \(=\) | \(\ds \map {\sigma_1} {343}\) | $\sigma_1$ of $343$ | |||||||||||
\(\ds 576\) | \(=\) | \(\ds \map {\sigma_1} {210}\) | $\sigma_1$ of $210$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {282}\) | $\sigma_1$ of $282$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {310}\) | $\sigma_1$ of $310$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {345}\) | $\sigma_1$ of $345$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {357}\) | $\sigma_1$ of $357$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {382}\) | $\sigma_1$ of $382$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {385}\) | $\sigma_1$ of $385$ | |||||||||||
\(\ds 784\) | \(=\) | \(\ds \map {\sigma_1} {364}\) | $\sigma_1$ of $364$ | |||||||||||
\(\ds 900\) | \(=\) | \(\ds \map {\sigma_1} {472}\) | $\sigma_1$ of $472$ | |||||||||||
\(\ds 961\) | \(=\) | \(\ds \map {\sigma_1} {400}\) | $\sigma_1$ of $400$ |
$\blacksquare$
Also see
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $36$