Ore Number/Examples/672

Example of Ore Number

$\map H {672} = 8$

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 $n$.

\(\ds \map {\sigma_0} {672}\)

\(=\)

\(\ds 24\)

$\sigma_0$ of $672$

\(\ds \map {\sigma_1} {672}\)

\(=\)

\(\ds 2016\)

$\sigma_1$ of $672$

\(\ds \leadsto \ \ \)

\(\ds \dfrac {672 \, \map {\sigma_0} {672} } {\map {\sigma_1} {672} }\)

\(=\)

\(\ds \dfrac {672 \times 24} {2016}\)

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {2^5 \times 3 \times 7} \times \paren {2^3 \times 3} } {\paren {2^5 \times 3^2 \times 7} }\)

\(\ds \)

\(=\)

\(\ds 8\)

$\blacksquare$