A number is called perfect if it is the sum of its proper divisors, i.e. all divisors less than itself. For example, 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.
Amicable numbers are a sort of generalization of perfect numbers. Two numbers a and b are said to be amicable if a is the sum of b‘s proper divisors and vice versa.
The ancient Greeks knew of only one pair of amicable numbers: 220 and 284. Medieval mathematician Thâbit ibn Kurrah discovered two more pairs: (17296, 18416) and (9363584, 9437056). Leonard Euler (1707–1783) found 58 more pairs. Now over 12 million amicable number pairs have been found.
To generalize things further, start with a number n and compute the sum of its proper divisors, then the sum of the divisors of that number, etc. This sequence of numbers is called the aliquot sequence of n. If this sequence is periodic, n is called a sociable number.
If the aliquot sequence has period 1, n is a perfect number. If the sequence has period 2, n is part of an amicable number pair.
Are there numbers whose aliquot sequence has period 3? Not that we know of. Currently the only aliquot sequence periods that have been demonstrated are 4, 5, 6, 8, 9, and 28.