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 $400$

$\blacksquare$


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Sources