Ore Number/Examples/1638

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Example of Ore Number

$H \left({1638}\right) = 9$

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 $n$.


\(\displaystyle \tau \left({1638}\right)\) \(=\) \(\displaystyle 24\) $\tau$ of $1638$
\(\displaystyle \sigma \left({1638}\right)\) \(=\) \(\displaystyle 4368\) $\sigma$ of $1638$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \dfrac {1638 \, \tau \left({1638}\right)} {\sigma \left({1638}\right)}\) \(=\) \(\displaystyle \dfrac {1638 \times 24} {4368}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\left({2 \times 3^2 \times 7 \times 13}\right) \times \left({2^3 \times 3}\right)} {\left({2^4 \times 3 \times 7 \times 13}\right)}\)
\(\displaystyle \) \(=\) \(\displaystyle 9\)

$\blacksquare$