Aliquot sum

Examples

Characterization of 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 n whose aliquot sums equal n + 1. The almost perfect numbers 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, who observed that both 2 and 5 are untouchable. 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.

  Iteration