Amicable number

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 credited them with many mystical properties.

A pair of amicable numbers constitutes an aliquot sequence of period 2.

A general formula by which these numbers could be derived was invented circa 850 by Thabit ibn Qurra (826-901): if

p = 3 × 2^n-1 - 1,

q = 3 × 2ⁿ - 1,

r = 9 × 2^2n-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 (17296, 18416) and the pair (9363584, 9437056). The pair (6232, 6368) are amicable, but they cannot be derived from this formula. In fact, this formula produces amicable numbers for n = 2, 4, and 7, but for no other values below 20000.

In every known case, the numbers of a pair are either both even or both odd. It is not known whether an even-odd pair of amicable numbers exist. Also, every known pair shares at least one common factor. It is not known whether a pair of coprime amicable numbers exist, though if they do, their product must be greater than 10⁶⁷. Also, a pair of coprime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar 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's formula was generalized by Euler.

If a number equals the sum of its own proper divisors, it is called a perfect number.