# Sum of Reciprocals of Divisors equals Abundancy Index

## Theorem

Let $n$ be a positive integer.

Let $\sigma \left({n}\right)$ be the sigma function of $n$.

Then:

$\displaystyle \sum_{d \mathop \backslash n} \frac 1 d = \frac {\sigma \left({n}\right)} n$

where $\dfrac {\sigma \left({n}\right)} n$ is the abundancy index of $n$.

## Proof

 $\displaystyle \sum_{d \mathop \backslash n} \frac 1 d$ $=$ $\displaystyle \sum_{d \mathop \backslash n} \frac 1 {\left({\frac n d}\right)}$ Sum Over Divisors Equals Sum Over Quotients $\displaystyle$ $=$ $\displaystyle \frac 1 n \sum_{d \mathop \backslash n} d$ $\displaystyle$ $=$ $\displaystyle \frac {\sigma \left({n}\right)} n$ Definition of Sigma Function

$\blacksquare$