Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive integer divisor other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) A pair of amicable numbers constitutes an aliquot sequence of period 2. A related concept is that of a perfect number, which is a number which equals the sum of its own proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable numbers.
For example, the smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.
The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368) (sequence A063990 in OEIS).
Read more about Amicable Numbers: History, Rules For Generation, Regular Pairs, Other Results, References in Popular Culture
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“Our religion vulgarly stands on numbers of believers. Whenever the appeal is made—no matter how indirectly—to numbers, proclamation is then and there made, that religion is not. He that finds God a sweet, enveloping presence, who shall dare to come in?”
—Ralph Waldo Emerson (1803–1882)