Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum

Theorem
Let $n \in \Z_{>0}$ be a positive integer.

The harmonic mean of the divisors of $n$ is given by:
 * $\map H n = \dfrac {n \, \map {\sigma_0} n} {\map {\sigma_1} n}$

where:
 * $\map {\sigma_0} n$ denotes the divisor count function: the number of divisors of $n$
 * $\map {\sigma_1} n$ denotes the divisor sum function: the sum of the divisors of $n$.