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