Amicable number

Two distinct positive integers $a,b$ are amicable when the sum of the proper divisors of $a$ is $b$ and the sum of the proper divisors of $b$ is $a$. The pair $220,284$ is an example. Iterating the proper-divisor sum can also produce longer directed cycles.

A pair of distinct positive integers in which each is the proper-divisor sum of the other. The integers $220$ and $284$ form an amicable pair.