Ratio of 360 to Aliquot Sum

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Theorem

$360$ has the property that its ratio to its aliquot sum is $4 : 9$.


Proof

The aliquot sum of an integer $n$ is the integer sum of the aliquot parts of $n$.

That is, the aliquot sum of $360$ is the divisor sum of $360$ minus $360$.

Thus:

\(\ds \map {\sigma_1} {360} - 360\) \(=\) \(\ds 1170 - 360\) $\sigma_1$ of $360$
\(\ds \) \(=\) \(\ds 810\)
\(\ds \leadsto \ \ \) \(\ds 4 \times \paren {\map {\sigma_1} {360} - 360}\) \(=\) \(\ds 3240\)
\(\ds \) \(=\) \(\ds 9 \times 360\)

$\blacksquare$


Also see


Historical Note

Leonard Eugene Dickson reports in his $1919$ work History of the Theory of Numbers, Volume I that this result was noted by Marin Mersenne.


Sources