Ratio of 2016 to Aliquot Sum
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Theorem
$2016$ 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 $2016$ is the divisor sum of $2016$ minus $2016$.
Thus:
| \(\ds \map {\sigma_1} {2016} - 2016\) | \(=\) | \(\ds 6552 - 2016\) | $\sigma_1$ of $2016$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 4536\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 4 \times \paren {\map {\sigma_1} {2016} - 2016}\) | \(=\) | \(\ds 18 \, 144\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 9 \times 2016\) |
$\blacksquare$
Historical Note
Leonard Eugene Dickson reports in his $1919$ work History of the Theory of Numbers, Volume I that this result was noted by Pierre de Fermat, in response to Marin Mersenne's observation that $360$ has the same property.
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