Amicable Pair/Examples/5020-5564

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Example of Amicable Pair

$5020$ and $5564$ are the $4$th amicable pair:

$\map \sigma {5020} = \map \sigma {5564} = 10 \, 584 = 5020 + 5564$


Proof

Let $\map s n$ denote the aliquot sum of $n$.

By definition:

$\map s n = \map \sigma n - n$

where $\map \sigma n$ denotes the $\sigma$ function.


Thus:

\(\displaystyle \map s {5020}\) \(=\) \(\displaystyle \map \sigma {5020} - 5020\)
\(\displaystyle \) \(=\) \(\displaystyle 10 \, 584 - 5020\) $\sigma$ of $5020$
\(\displaystyle \) \(=\) \(\displaystyle 5564\)


\(\displaystyle \map s {5564}\) \(=\) \(\displaystyle \map \sigma {5564} - 5564\)
\(\displaystyle \) \(=\) \(\displaystyle 5544 - 5564\) $\sigma$ of $5564$
\(\displaystyle \) \(=\) \(\displaystyle 10 \, 584\)

$\blacksquare$


Sources