Ore Number/Examples/140
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Example of Ore Number
- $\map H {140} = 5$
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 $n$.
\(\ds \map {\sigma_0} {140}\) | \(=\) | \(\ds 12\) | $\sigma_0$ of $140$ | |||||||||||
\(\ds \map {\sigma_1} {140}\) | \(=\) | \(\ds 336\) | $\sigma_1$ of $140$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {140 \, \map {\sigma_0} {140} } {\map {\sigma_1} {140} }\) | \(=\) | \(\ds \dfrac {140 \times 12} {336}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2^2 \times 5 \times 7} \times \paren {2^2 \times 3} } {\paren {2^4 \times 3 \times 7} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5\) |
$\blacksquare$