Aliquot sum

In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of, that is, all divisors of other than 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 are, and 6, so the aliquot sum of 12 is 16 i.e..
The values of for are:

The aliquot sum function 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 are the numbers whose aliquot sums equal. The almost perfect numbers are the numbers whose aliquot sums equal.
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, who observed that both 2 and 5 are untouchable. Paul 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, the aliquot sum is.

The mathematicians noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.

the aliquot sum function produces the aliquot sequence of a nonnegative integer .
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.