# Ore Number/Examples/1638

## Example of Ore Number

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

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 $n$.

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

$\blacksquare$