# Amicable Pair/Examples/5020-5564

## 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:

 $\ds \map s {5020}$ $=$ $\ds \map \sigma {5020} - 5020$ $\ds$ $=$ $\ds 10 \, 584 - 5020$ $\sigma$ of $5020$ $\ds$ $=$ $\ds 5564$

 $\ds \map s {5564}$ $=$ $\ds \map \sigma {5564} - 5564$ $\ds$ $=$ $\ds 5544 - 5564$ $\sigma$ of $5564$ $\ds$ $=$ $\ds 10 \, 584$

$\blacksquare$