Ore Number/Examples/140

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Example of Ore Number

$H \left({140}\right) = 5$

where $H \left({n}\right)$ denotes the harmonic mean of the divisors of $n$.


Proof

From Harmonic Mean of Divisors in terms of Tau and Sigma:

$H \left({n}\right) = \dfrac {n \, \tau \left({n}\right)} {\sigma \left({n}\right)}$

where:

$\tau \left({n}\right)$ denotes the $\tau$ (tau) function: the number of divisors of $n$
$\sigma \left({n}\right)$ denotes the $\sigma$ (sigma) function: the sum of the divisors of $n$.


\(\displaystyle \tau \left({140}\right)\) \(=\) \(\displaystyle 12\) $\tau$ of $140$
\(\displaystyle \sigma \left({140}\right)\) \(=\) \(\displaystyle 336\) $\sigma$ of $140$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \dfrac {140 \, \tau \left({140}\right)} {\sigma \left({140}\right)}\) \(=\) \(\displaystyle \dfrac {140 \times 12} {336}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\left({2^2 \times 5 \times 7}\right) \times \left({2^2 \times 3}\right)} {\left({2^4 \times 3 \times 7}\right)}\)
\(\displaystyle \) \(=\) \(\displaystyle 5\)

$\blacksquare$