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.

## Contents

## 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.
^{[1]} - The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively.
^{[1]}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.
^{[1]}^{[2]}Erdős proved that their number is infinite.^{[3]}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.^{[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 do not exist).^{[1]}

## References

- ^
^{a}^{b}^{c}^{d}^{e}Pollack, Paul; Pomerance, Carl (2016), "Some problems of Erdős on the sum-of-divisors function",*Transactions of the American Mathematical Society*, Series B,**3**: 1–26, doi:10.1090/btran/10, MR 3481968 **^**Sesiano, J. (1991), "Two problems of number theory in Islamic times",*Archive for History of Exact Sciences*,**41**(3): 235–238, JSTOR 41133889, MR 1107382**^**Erdős, P. (1973), "Über die Zahlen der Form und " (PDF),*Elemente der Mathematik*,**28**: 83–86, MR 0337733