Aliquot sum

The aliquot sum of a positive integer is the sum of its positive divisors other than itself. It is commonly written s(n).

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Proper divisor: A positive divisor of an integer that is smaller than the integer itself. Aliquot sums add precisely these divisors.
Perfect number: A positive integer equal to the sum of its positive proper divisors. A perfect number is a fixed point of the aliquot-sum operation.
Euclid: An ancient Greek mathematician whose surviving works include the Elements. Euclid gave a formula for even perfect numbers using Mersenne primes.
Abundant number: A positive integer whose proper divisors sum to more than the integer. Its aliquot sum exceeds the input, unlike the deficient case.
Prime number: A whole number greater than one with exactly two positive divisors: one and itself. Every prime has aliquot sum one because its only proper divisor is one.
Amicable numbers: A pair of distinct integers whose proper-divisor sums equal each other’s values. Each member’s aliquot sum is the other member.
Mersenne prime: A prime number of the form 2^p−1, where p is itself prime. Mersenne primes connect Euclid’s construction to the aliquot sums of perfect numbers.
Deficient number: A positive integer whose proper divisors sum to less than the integer. Its aliquot sum falls below the input, the opposite inequality from abundance.
Divisor function: A function that assigns arithmetic information about the divisors of an integer. The sum-of-divisors function gives the aliquot sum after subtracting the integer.
Sociable number: An integer in a cycle of length greater than two under repeated proper-divisor summation. Its aliquot-sum sequence returns to its starting value after several steps.