Aliquot sum

Characterization of classes of numbers

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 (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. 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 pq, the aliquot sum is p + q + 1.

The mathematicians Pollack & Pomerance (2016) 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.