Ore Number/Examples/140

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Example of Ore Number

$\map H {140} = 5$

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 $n$.


\(\ds \map {\sigma_0} {140}\) \(=\) \(\ds 12\) $\sigma_0$ of $140$
\(\ds \map {\sigma_1} {140}\) \(=\) \(\ds 336\) $\sigma_1$ of $140$
\(\ds \leadsto \ \ \) \(\ds \dfrac {140 \, \map {\sigma_0} {140} } {\map {\sigma_1} {140} }\) \(=\) \(\ds \dfrac {140 \times 12} {336}\)
\(\ds \) \(=\) \(\ds \dfrac {\paren {2^2 \times 5 \times 7} \times \paren {2^2 \times 3} } {\paren {2^4 \times 3 \times 7} }\)
\(\ds \) \(=\) \(\ds 5\)

$\blacksquare$