For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6).
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.[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]Paul 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 numberpq, the aliquot sum is p + q + 1.[1]
The mathematicians
Pollack & Pomerance (2016) noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
^
abcdPollack, 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,
MR3481968