Mathematical Foundations:
Definition: Perfect Number
A positive integer n is perfect if the sum of its proper divisors (divisors excluding n itself) is equal to n. Example: 28 is perfect, as 1 + 2 + 4 + 7 + 14 = 28.
Definition: Deficient and Abundant Numbers
A positive integer n is deficient if the sum of its proper divisors is strictly less than n. It is abundant if the sum of its proper divisors is strictly greater than n.
Properties of Abundant Numbers:
- 12 is the smallest abundant number (1 + 2 + 3 + 4 + 6 = 16 > 12).
- 24 is the smallest positive integer that can be expressed as the sum of two abundant numbers.
- Theorem: All integers greater than 28123 can be expressed as the sum of two abundant numbers.
- Note: The upper bound of 28123, derived from mathematical analysis, is not a tight bound. The greatest integer that cannot be represented as the sum of two abundant numbers is strictly less than 28123.
⎯ Adapted from Project Euler Problem 23, CC BY-NC-SA 4.0
Problem Statement:
Determine the sum of all positive integers that cannot be represented as the sum of two abundant numbers.