Abundant Number Checker

About this calculator

A number is called abundant (or excessive) when the sum of its proper divisors — every divisor except the number itself — exceeds the number: 12 is the smallest example, since its proper divisors 1, 2, 3, 4, and 6 add up to 16. This classification sits alongside perfect numbers, whose proper divisors sum to exactly the number (like 6 = 1 + 2 + 3), and deficient numbers, whose proper divisors sum to less than the number — a three-way split of the integers first studied systematically by the ancient Greek mathematician Nicomachus of Gerasa around 100 CE.

The concept lives mainly in number theory and recreational mathematics today: it underlies results like the fact that every multiple of an abundant number is itself abundant, and it appears in classroom introductions to divisor functions and the sigma function used to compute the sum of divisors.

This calculator takes a whole number, lists its proper divisors, sums them, and tells you whether the number is abundant, deficient, or perfect.