In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself.
The values of s(n) for n = 1, 2, 3, ... are: The aliquot sum function can be used to characterize several notable classes of numbers: The mathematicians Pollack & Pomerance (2016) noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
Iterating the aliquot sum function produces the aliquot sequence n, s(n), s(s(n)), … of a nonnegative integer n (in this sequence, we define s(0) = 0).
Sociable numbers are numbers whose aliquot sequence is a periodic sequence.
It remains unknown whether these sequences always end with a prime number, a perfect number, or a periodic sequence of sociable numbers.