Ore Number/Examples/1638
Jump to navigation
Jump to search
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 $1638$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {1638 \, \map {\sigma_0} {1638} } {\map {\sigma_1} {1638} }\) | \(=\) | \(\ds \dfrac {1638 \times 24} {4368}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {2 \times 3^2 \times 7 \times 13} \times \paren {2^3 \times 3} } {\paren {2^4 \times 3 \times 7 \times 13} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9\) |
$\blacksquare$