Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. Such a pair 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. 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 circa 850 by Thabit ibn Qurra (826-901): if
- p = 3 × 2n-1 - 1,
- q = 3 × 2n - 1,
- 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 amicable pair (220, 284), as well as the pair (17,296, 18,416) and the pair (9,363,584, 9,437,056). The pair (6232, 6368) are amicable, but they cannot be derived from this formula.
Amicable numbers have been studied by Al Madshritti (died 1007), Abu Mansur Tahir al-Baghdadi (980-1037), René Descartes (1596-1650), to whom the formula of Thabit is sometimes ascribed, C. Rudolphus and others. Thabit formula was generalized by Euler.
If a number equals the sum of its own proper divisors, it is called a perfect number.