Sum of Reciprocals of Divisors of Perfect Number is 2
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Theorem
Let $n$ be a perfect number.
Then:
- $\ds \sum_{d \mathop \divides n} \dfrac 1 d = 2$
That is, the sum of the reciprocals of the divisors of $n$ is equal to $2$.
Proof
\(\ds \sum_{d \mathop \divides n} d\) | \(=\) | \(\ds \map {\sigma_1} n\) | Definition of Divisor Sum Function | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac 1 n \sum_{d \mathop \divides n} d\) | \(=\) | \(\ds \dfrac {\map {\sigma_1} n} n\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{d \mathop \divides n} \frac d n\) | \(=\) | \(\ds \dfrac {\map {\sigma_1} n} n\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{d \mathop \divides n} \frac 1 d\) | \(=\) | \(\ds \dfrac {\map {\sigma_1} n} n\) |
The result follows by definition of perfect number:
A perfect number $n$ is a (strictly) positive integer such that:
- $\dfrac {\map {\sigma_1} n} n = 2$
where $\sigma_1: \Z_{>0} \to \Z_{>0}$ is the divisor sum function.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $28$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $28$