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. That is, It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.

Examples

For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6). The values of s(n) for n = 1, 2, 3, ... are:

Characterization of classes of numbers

The aliquot sum function can be used to characterize several notable classes of numbers: n + 1 . The almost perfect numbers (which include the powers of 2, being the only known such numbers so far) are the numbers n whose aliquot sums equal n – 1 . p + q + 1 . The mathematicians noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.

Iteration

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. Amicable numbers are sociable numbers whose aliquot sequence has period 2. It remains unknown whether these sequences always end with a prime number, a perfect number, or a periodic sequence of sociable numbers.