Sum of Reciprocals of Divisors equals Abundancy Index

Theorem
Let $n$ be a positive integer.

Let $\map \sigma n$ denote the sigma function of $n$.

Then:
 * $\displaystyle \sum_{d \mathop \divides n} \frac 1 d = \frac {\map \sigma n} n$

where $\dfrac {\map \sigma n} n$ is the abundancy index of $n$.