As an exercise in Python programming I've written a script that finds amicable pairs, and I've had fun doing so. But I'm wondering if amicable pairs have a use, in mathematics or elsewhere?
As a reminder:
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive integer divisor other than the number itself. For example, the proper divisors of $6$ are $1$, $2$, and $3$.)
For example, the smallest pair of amicable numbers is $(220, 284)$; for the proper divisors of $220$ are $1$, $2$, $4$, $5$, $10$, $11$, $20$, $22$, $44$, $55$ and $110$, of which the sum is $284$; and the proper divisors of $284$ are $1$, $2$, $4$, $71$, and $142$, of which the sum is $220$. (from the Wikipedia article)
The first few amicable pairs, according to Math World, are: $(220, 284)$, $(1184, 1210)$, $(2620, 2924)$, $(5020, 5564)$, $(6232, 6368)$, $(10744, 10856)$, $(12285, 14595)$, $(17296, 18416)$, $(63020, 76084)$