Amicable numbers are two numbers so related that the sum of the divisors of the one is equal to the other, unity being considered as a divisor but not the number itself. Such a pair are 220 and 284; for the divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220. Amicable numbers were known to the Pythagoreans, who accredited them with many mystical properties. A general formula by which these numbers could be derived was invented by the Arabian astronomer Thabit ibn Kurrah (836-901): if p = 3 × 2n-1 - 1, q = 3 × 2n - 1 and r = 9 × 22n-1 - 1, where n > 1 is an integer and p, q, and r are prime numbers, then 2npq and 2nr are a pair of amicable numbers. This formula gives the pairs 220 and 284, 17,296 and 18,416, 9,463,584 and 9,437,056. The pair 6232 and 6368 are amicable, but they cannot be derived from this formula. Amicable numbers have been studied by Al Madshritti (d. 1007), Rene Descartes, to whom the formula of Thabit ibn Kurrah is sometimes ascribed, C. Rudolphus and others.
A pair of amicable numbers equal the sum of each other's divisors. If a number equals the sum of its own divisors, it is called a perfect number.
- (from an old encyclopedia)