Ore Number/Examples/672

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Example of Ore Number

$H \left({672}\right) = 8$

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 $n$.


\(\displaystyle \tau \left({672}\right)\) \(=\) \(\displaystyle 24\) $\tau$ of $672$
\(\displaystyle \sigma \left({672}\right)\) \(=\) \(\displaystyle 2016\) $\sigma$ of $672$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \dfrac {672 \, \tau \left({672}\right)} {\sigma \left({672}\right)}\) \(=\) \(\displaystyle \dfrac {672 \times 24} {2016}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\left({2^5 \times 3 \times 7}\right) \times \left({2^3 \times 3}\right)} {\left({2^5 \times 3^2 \times 7}\right)}\)
\(\displaystyle \) \(=\) \(\displaystyle 8\)

$\blacksquare$