Two numbers M and N are called “amicable” (friendly) if each equals the sum of the proper divisors of the other number. (The proper divisors of a natural number do not include the number itself.) M = 220 and N = 284 form a pair of amicable numbers.
Proper divisors of 220 = (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110)
Proper divisors of 284 = (1, 2, 4, 71, 142)
220 = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
284 = 1 + 2 + 4 + 71 + 142 = 220
The Arab mathematician Abu’l-Hasan Thābit ibn Qurra al-Ḥarrānī (830–901) proved:
If p = 3·2n − 1, q = 3·2n−1 − 1, r = 9·22n−1 − 1 (n = 2, 3, 4, 5, …) are primes, then M = 2n·p·q and N = 2n·r are amicable numbers.
n = 2, p = 11, q = 5, r = 71, M = 22·11·5 = 220, N = 22·71 = 284
n = 3, M and N do not exist
n = 4, p = 47, q = 23, r = 1151, M = 24·47·23 = 17 296, N = 24·1151 = 18 416
n = 7, M = 9 363 584, N = 9 437 056