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A pair (a, b) of positive integers is said to be amicable if a is the sum of the divisors of b, and b is the sum of the divisors of a. The smallest such pair is (284, 220).

Amicable (another word for friendly) numbers were first found (circa 600 BC) by the ancient Greeks, and the most famous one, (220, 284) was respected as a symbol of friendship by the Pythagoreans. Like perfect numbers, amicable numbers are determined by their proper divisors -- numbers they are divisible by other than themselves. Whereas a perfect number's divisors add up to equal that same number, amicable number divisors add up to equal another number, which has divisors that add up to the original. Thus they form a looped, two step aliquot sequence. While as of yet they have no practical applications in mathematics, lots of study has gone into these two step aliquot cycles, and over half a million of them are now known.

Around 850 AD, Thabit ibn Kurrah proved that for n›1, if p=3.2n-1-1, q=3.2n-1, and r=9.22n-1-1 are prime, then 2npq and 2nr are amicable numbers. This formula was rediscovered and used, hundreds of years later, by Fermat and Descartes to find the pairs (17296,18416) with n=3 and (9363584,9437056) with n=7 respectively. Fermat's work with amicable numbers also lead to his discovery of Fermat's Little Theorem. Around 1640, Euler outdid them both using modified formulas to find a list of 64 amicable numbers, although five of them were later found to be not amicable. In 1866, a sixteen year old named Nicolo Paganini (note, not the same person as Niccolo Paganini, violin virtuoso) discovered the second lowest valued pair, (1184,1210), through trial and error.

It is assumed, though it has not yet been proven, that there exist infinitely many pairs of amicable numbers. There are odd pairs such as (12285,14595) and (67095,71145), but it isn't known if there exists a pair where one number is even and the other odd. L.E. Dickson and T. E. Mason have found amicable triplets, which form aliquot cycles of length three -- examples are (1980,2016,2556) and (103340640,123228768,124015008).

There's a list of known amicable pairs, complete up to lengths of 1012, at http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htm

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