amicable numbers

Definition.

Two natural numbers $n$ and $m$ are amicable numbers (an amicable pair or simply amicable) if each is the sum of the proper divisors of the other.

Example.

The smallest amicable pair is $n=220$ and $m=284$. Indeed the sum of proper divisors of $220$ is:

$$1+2+4+5+10+11+20+22+44+55+110=284$$

and the sum of proper divisors of $284$ is:

$$1+2+4+71+142=220.$$