Ore Number/Examples/1638

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Example of Ore Number

$\map H {1638} = 9$

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 $n$.


\(\ds \map {\sigma_0} {1638}\) \(=\) \(\ds 24\) $\sigma_0$ of $1638$
\(\ds \map {\sigma_1} {1638}\) \(=\) \(\ds 4368\) $\sigma_1$ of $1638$
\(\ds \leadsto \ \ \) \(\ds \dfrac {1638 \, \map {\sigma_0} {1638} } {\map {\sigma_1} {1638} }\) \(=\) \(\ds \dfrac {1638 \times 24} {4368}\)
\(\ds \) \(=\) \(\ds \dfrac {\paren {2 \times 3^2 \times 7 \times 13} \times \paren {2^3 \times 3} } {\paren {2^4 \times 3 \times 7 \times 13} }\)
\(\ds \) \(=\) \(\ds 9\)

$\blacksquare$