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

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 $270$

\(\displaystyle \leadsto \ \ \)

\(\displaystyle \dfrac {270 \, \tau \left({270}\right)} {\sigma \left({270}\right)}\)

\(=\)

\(\displaystyle \dfrac {270 \times 16} {720}\)

\(\displaystyle \)

\(=\)

\(\displaystyle \dfrac {\left({2 \times 3^3 \times 5}\right) \times 2^4} {\left({2^4 \times 3^2 \times 5}\right)}\)

\(\displaystyle \)

\(=\)

\(\displaystyle 6\)

$\blacksquare$