Ore Number/Examples/28

Example of Ore Number

$\map H {28} = 3$

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 $n$.

\(\ds \map {\sigma_0} {28}\)

\(=\)

\(\ds 6\)

$\sigma_0$ of $28$

\(\ds \map {\sigma_1} {28}\)

\(=\)

\(\ds 56\)

$\sigma_1$ of $28$

\(\ds \leadsto \ \ \)

\(\ds \dfrac {28 \map {\sigma_0} {28} } {\map {\sigma_1} {28} }\)

\(=\)

\(\ds \dfrac {28 \times 6} {56}\)

\(\ds \)

\(=\)

\(\ds 3\)

$\blacksquare$