Ore Number/Examples/270

Example of Ore Number

$\map H {270} = 6$

where $\map H n$ denotes the harmonic mean of the divisors of $n$.

Proof

From Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum:

$\map H n = \dfrac {n \, \map {\sigma_0} n} {\map {\sigma_1} n}$

where:

$\map {\sigma_0} n$ denotes the divisor count function: the number of divisors of $n$

$\map {\sigma_1} n$ denotes the divisor sum function: the sum of the divisors of $n$.

\(\ds \map {\sigma_0} {270}\)

\(=\)

\(\ds 16\)

$\sigma_0$ of $270$

\(\ds \map {\sigma_1} {270}\)

\(=\)

\(\ds 720\)

$\sigma_1$ of $270$

\(\ds \leadsto \ \ \)

\(\ds \dfrac {270 \, \map {\sigma_0} {270} } {\map {\sigma_1} {270} }\)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {2 \times 3^3 \times 5} \times 2^4} {\paren {2^4 \times 3^2 \times 5} }\)

\(\ds \)

\(=\)

\(\ds 6\)

$\blacksquare$