Ore Number/Examples/6

Example of Ore Number

$\map H 6 = 2$

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 $n$.

\(\ds \map {\sigma_0} 6\)

\(=\)

\(\ds 4\)

$\sigma_0$ of $6$

\(\ds \map {\sigma_1} 6\)

\(=\)

\(\ds 12\)

$\sigma_1$ of $6$

\(\ds \leadsto \ \ \)

\(\ds \dfrac {6 \, \map {\sigma_0} 6} {\map {\sigma_1} 6}\)

\(=\)

\(\ds \dfrac {6 \times 4} {12}\)

\(\ds \)

\(=\)

\(\ds 2\)

$\blacksquare$