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
- 1919: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { I }$: Chapter $\text {II}$. Formulas for the Number and Sum of Divisors, Problems of Fermat and Wallis