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

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 $1638$

\(\displaystyle \leadsto \ \ \)

\(\displaystyle \dfrac {1638 \, \tau \left({1638}\right)} {\sigma \left({1638}\right)}\)

\(=\)

\(\displaystyle \dfrac {1638 \times 24} {4368}\)

\(\displaystyle \)

\(=\)

\(\displaystyle \dfrac {\left({2 \times 3^2 \times 7 \times 13}\right) \times \left({2^3 \times 3}\right)} {\left({2^4 \times 3 \times 7 \times 13}\right)}\)

\(\displaystyle \)

\(=\)

\(\displaystyle 9\)

$\blacksquare$