Propersum

Propersum is a term used in number theory to denote the sum of the proper divisors of a positive integer. It is commonly represented as propersum(n) or s(n) and equals the sum of all positive divisors of n except n itself. Equivalently, propersum(n) = sigma(n) - n, where sigma is the divisor-sum function. By convention, propersum(1) is taken to be 0.

Examples illustrate the concept: propersum(6) = 1 + 2 + 3 = 6, propersum(12) = 1 + 2 + 3 + 4 + 6 = 16, and propersum(28) = 1 + 2 + 4 + 7 + 14 = 28. These examples show how propersum can equal, exceed, or fall short of the original number.

Classification linked to propersum includes perfect, abundant, and deficient numbers. A number n is perfect if propersum(n) = n (e.g., 6, 28, 496). If propersum(n) > n, n is abundant; if propersum(n) < n, n is deficient. Amicable numbers are distinct pairs (a, b) such that propersum(a) = b and propersum(b) = a, with a ≠ b (famously 220 and 284).

Key properties include the relationship propersum(n) = sigma(n) - n. While sigma(n) is multiplicative, propersum(n) is not in general. For a prime power p^k, propersum(p^k) = (p^{k+1} - 1)/(p - 1) - p^k.

Computationally, propersum(n) is typically found by enumerating divisors up to sqrt(n) and adding both divisors when they divide n, yielding a time complexity around O(sqrt(n)) per calculation. See also divisor functions, perfect numbers, deficient and abundant numbers, and amicable numbers.