# Ore Number/Examples/270

## Example of Ore Number

$H \left({270}\right) = 6$

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

## Proof

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

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 \ \$ $\displaystyle \dfrac {270 \, \tau \left({270}\right)} {\sigma \left({270}\right)}$ $=$ $\displaystyle \dfrac {270 \times 16} {720}$ $\displaystyle$ $=$ $\displaystyle \dfrac {\left({2 \times 3^3 \times 5}\right) \times 2^4} {\left({2^4 \times 3^2 \times 5}\right)}$ $\displaystyle$ $=$ $\displaystyle 6$

$\blacksquare$