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

Example of Odd Amicable Pair

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

$\map {\sigma_1} {12 \, 285} = \map {\sigma_1} {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$

Proof

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

$\map {\sigma_1} m = \map {\sigma_1} n = m + n$

where $\map {\sigma_1} n$ denotes the divisor sum function.

Thus:

$\ds \map {\sigma_1} {12 \, 285}$

$=$

$\ds 26 \, 880$

$\sigma_1$ of $12 \, 285$

$\ds $

$=$

$\ds 12 \, 285 + 14 \, 595$

$\ds $

$=$

$\ds \map {\sigma_1} {14 \, 595}$

$\sigma_1$ 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

Jun. - Jul. 1939: B.H. Brown: A New Pair of Amicable Numbers (Amer. Math. Monthly Vol. 46, no. 6: p. 345) www.jstor.org/stable/2302890

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12,285$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12,285$