Ore Number/Examples/496

Example of Ore Number

$\map H {496} = 5$

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 $n$.

\(\ds \map {\sigma_0} {496}\)

\(=\)

\(\ds 10\)

$\sigma_0$ of $496$

\(\ds \map {\sigma_1} {496}\)

\(=\)

\(\ds 992\)

$\sigma_1$ of $496$

\(\ds \leadsto \ \ \)

\(\ds \dfrac {496 \, \map {\sigma_0} {496} } {\map {\sigma_1} {496} }\)

\(=\)

\(\ds \dfrac {496 \times 10} {992}\)

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {2^4 \times 31} \times \paren {2 \times 5} } {\paren {2^5 \times 31} }\)

\(\ds \)

\(=\)

\(\ds 5\)

$\blacksquare$