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
- 1948: Øystein Ore: On the averages of the divisors of a number (Amer. Math. Monthly Vol. 55, no. 10: pp. 615 – 619) www.jstor.org/stable/2305616