Perfect Number Checker
Determine whether a number is perfect, abundant, or deficient by summing its proper divisors. Use it when studying number theory, checking homework, or exploring classic mathematical puzzles.
About this calculator
A proper divisor of n is any positive integer less than n that divides n evenly. The sum of proper divisors is called the aliquot sum, σ(n) − n, where σ(n) is the standard divisor function. Three classifications exist: if the aliquot sum equals n the number is Perfect (e.g., 6 = 1+2+3); if it exceeds n the number is Abundant (e.g., 12, aliquot sum = 16); if it falls short the number is Deficient (e.g., 8, aliquot sum = 7). The formula used here iterates i from 1 to n−1, collects every i where n % i = 0, and sums them. Perfect numbers are extraordinarily rare — only five are known below 10^18 — making this an enduring mystery in mathematics.
How to use
Enter the number 28 and select 'single number' analysis. The calculator iterates from 1 to 27, finding divisors: 1, 2, 4, 7, 14. It sums them: 1 + 2 + 4 + 7 + 14 = 28. Because the aliquot sum equals the number itself (28 = 28), the result is Perfect. Now try 12: divisors are 1, 2, 3, 4, 6 → sum = 16. Since 16 > 12, the calculator classifies 12 as Abundant.
Frequently asked questions
What makes a number a perfect number and how rare are they?
A perfect number equals the sum of all its proper divisors. The first four perfect numbers are 6, 28, 496, and 8128, discovered in antiquity. The next known perfect number is 33,550,336. All known perfect numbers are even, and it remains an open question in mathematics whether any odd perfect numbers exist. Their rarity makes them a fascinating topic in recreational and professional number theory alike.
What is the difference between an abundant number and a deficient number?
An abundant number has an aliquot sum greater than itself — it has 'more than enough' divisors. The smallest abundant number is 12 (divisor sum = 16). A deficient number has an aliquot sum less than itself, meaning its divisors fall short; all prime numbers are deficient because their only proper divisor is 1. The vast majority of positive integers are deficient, while abundant numbers become more common as numbers grow larger.
How can I use this calculator to find all perfect numbers up to a limit?
Enable the range check mode and enter an upper limit, for example 10,000. The calculator will test every integer from 2 to your limit, applying the aliquot sum formula to each, and return a list of all perfect numbers found. This is useful for math courses, competitive programming practice, or simply exploring number-theory patterns without writing your own code.