Two numbers are amicable when each one equals the sum of the other's proper divisors. The classic pair is 220 and 284: the proper divisors of 220 (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110) sum to 284, and the proper divisors of 284 (1, 2, 4, 71, 142) sum to 220. They are perfect numbers' sociable cousins — where a perfect number is its own aliquot sum, amicable numbers point at each other.
The pair (220, 284) was known to the Pythagoreans, who treated it as a symbol of friendship, and it recurs in mysticism and medieval talismans. For centuries it was the only known pair until Thābit ibn Qurra (9th century) gave a formula generating others; Fermat, Descartes, and Euler later rediscovered and extended the hunt — Euler alone found 58 pairs. Remarkably, the small pair 1184 & 1210 was missed by all of them and only found in 1866 by the 16-year-old Nicolò Paganini.
Amicable numbers are the length-2 cycles of the aliquot sequence (iterating "sum of proper divisors"). Longer cycles are called sociable numbers. It is unknown whether infinitely many amicable pairs exist, or whether any pair consists of one odd and one even number.