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