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

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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 $n$.


Proof

\(\ds \frac 1 {\map H n}\) \(=\) \(\ds \frac 1 {\map {\sigma_0} n} \paren {\sum_{d \mathop \divides n} \frac 1 d}\) Definition of Harmonic Mean
\(\ds \sum_{d \mathop \divides n} \frac 1 d\) \(=\) \(\ds \frac {\map {\sigma_1} n} n\) Sum of Reciprocals of Divisors equals Abundancy Index
\(\ds \leadsto \ \ \) \(\ds \frac 1 {\map H n}\) \(=\) \(\ds \frac {\map {\sigma_1} n} {n \, \map {\sigma_0} n}\)
\(\ds \leadsto \ \ \) \(\ds \map H n\) \(=\) \(\ds \frac {n \, \map {\sigma_0} n} {\map {\sigma_1} n}\)

$\blacksquare$


Sources

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