Ore Number/Examples/28
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Example of Ore Number
- $\map H {28} = 3$
where $\map H n$ denotes the harmonic mean of the divisors of $n$.
Proof
From Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum:
- $\map H n = \dfrac {n \map {\sigma_0} n} {\map {\sigma_1} n}$
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 $n$.
\(\ds \map {\sigma_0} {28}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $28$ | |||||||||||
\(\ds \map {\sigma_1} {28}\) | \(=\) | \(\ds 56\) | $\sigma_1$ of $28$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {28 \map {\sigma_0} {28} } {\map {\sigma_1} {28} }\) | \(=\) | \(\ds \dfrac {28 \times 6} {56}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 3\) |
$\blacksquare$