# Odd Amicable Pair/Examples/1,175,265-1,438,983

## Example of Odd Amicable Pair

$1 \, 175 \, 265$ and $1 \, 438 \, 983$ are the $9$th odd amicable pair:

$\map \sigma {1 \, 175 \, 265} = \map \sigma {1 \, 438 \, 983} = 2 \, 614 \, 240 = 1 \, 175 \, 265 + 1 \, 438 \, 983$

## Proof

By definition, $m$ and $n$ form an amicable pair if and only if:

$\map \sigma m = \map \sigma n = m + n$

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

Thus:

 $\ds \map \sigma {1 \, 175 \, 265}$ $=$ $\ds 2 \, 614 \, 240$ $\sigma$ of $1 \, 175 \, 265$ $\ds$ $=$ $\ds 1 \, 175 \, 265 + 1 \, 438 \, 983$ $\ds$ $=$ $\ds \map \sigma {1 \, 438 \, 983}$ $\sigma$ of $1 \, 438 \, 983$

$\blacksquare$

## Historical Note

The odd amicable pair $1 \, 175 \, 265$ and $1 \, 438 \, 983$ was discovered by G.W. Kraft in the $17$th century.

It was the $1$st odd amicable pair to be discovered.