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:
Characterization of classes of numbers.
 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:
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).