# Aliquot sum

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

## Examples

For example, the proper divisors of 15 (that is, the positive divisors of 15 that are not equal to 15) are 1, 3 and 5, so the aliquot sum of 15 is 9 i.e. (1 + 3 + 5).

The values of s(n) for n = 1, 2, 3, ... are:

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ... (sequence A001065 in the OEIS)

## Characterization of classes of numbers

Pollack & Pomerance (2016) write that the aliquot sum function was one of Paul Erdős's "favorite subjects of investigation". It can be used to characterize several notable classes of numbers:

• 1 is the only number whose aliquot sum is 0. A number is prime if and only if its aliquot sum is 1.
• The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively. The quasiperfect numbers (if such numbers exist) are the numbers n whose aliquot sums equal 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.
• The untouchable numbers are the numbers that are not the aliquot sum of any other number. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable. Erdős proved that their number is infinite. The conjecture that 5 is the only odd untouchable number remains unproven, but would follow from a form of Goldbach's conjecture together with the observation that, for a semiprime number pq, the aliquot sum is p + q + 1.

## 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). It remains unknown whether these sequences always converge (the limit of the sequence must be 0 or a perfect number), or whether they can diverge (i.e. the limit of the sequence does not exist).