Ore Number/Examples/270
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Example of Ore Number
- $H \left({270}\right) = 6$
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 $n$.
\(\displaystyle \tau \left({270}\right)\) | \(=\) | \(\displaystyle 16\) | $\tau$ of $270$ | ||||||||||
\(\displaystyle \sigma \left({270}\right)\) | \(=\) | \(\displaystyle 720\) | $\sigma$ of $270$ | ||||||||||
\(\displaystyle \leadsto \ \ \) | \(\displaystyle \dfrac {270 \, \tau \left({270}\right)} {\sigma \left({270}\right)}\) | \(=\) | \(\displaystyle \dfrac {270 \times 16} {720}\) | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {\left({2 \times 3^3 \times 5}\right) \times 2^4} {\left({2^4 \times 3^2 \times 5}\right)}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle 6\) |
$\blacksquare$