Odd Amicable Pair/Examples/12,285-14,595

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Example of Odd Amicable Pair

$12 \, 285$ and $14 \, 595$ are the $7$th amicable pair and the smallest odd amicable pair:

$\map \sigma {12 \, 285} = \map \sigma {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$


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:

\(\displaystyle \\map \sigma {12 \, 285}\) \(=\) \(\displaystyle 26 \, 880\) $\sigma$ of $12 \, 285$
\(\displaystyle \) \(=\) \(\displaystyle 12 \, 285 + 14 \, 595\)
\(\displaystyle \) \(=\) \(\displaystyle \map \sigma {14 \, 595}\) $\sigma$ of $14 \, 595$


It can be determined by inspection of the aliquot sums of all smaller odd integers that there is no smaller odd amicable pair.

$\blacksquare$


Historical Note

The odd amicable pair $12 \, 285$ and $14 \, 595$ was discovered by B.H. Brown, who reported on them in the American Mathematical Monthly in $1939$.


Sources