A pair (p, q) of numbers for which the sum of the proper divisors of p is equal to q and the sum of the proper divisors of q is equal to p.
Amicable numbers are sometimes called sociable numbers.
There do not seem to be any amicable numbers formed by an even number and an odd number.
Examples
Here are a few pairs of amicable numbers: (1184, 1210), (2620, 2924), (5020, 5564) and (6232, 6368).
The least pair of amicable numbers is (220, 284).
- The set of proper divisors of 220 is equal to 284:
divp(220) = {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110} and the sum of these numbers is 284. - The set of proper divisors of 284 is equal to 220:
divp(284) = {1, 2, 4, 71, 142} and the sum of these numbers is 220.
Historical Note
The origins of mathematicians’ interest in amicable numbers is lost in Antiquity when these numbers were attributed magical properties. We believe that the Pythagoreans knew these numbers and associated certain social qualities with them. Pythagoras talked about a friend that was “another I” like the numbers 220 and 284.