Perfect Number Checker
Determine whether a number equals the sum of its proper divisors.
Inputs
Positive integer to test for perfection.
Is Perfect?
Yes
Sum of Proper Divisors
28
Classification
Perfect
Proper Divisors
1, 2, 4, 7, 14
Step by step
Proper divisors
1 + 2 + 4 + 7 + 14
Sum of proper divisors
= 28
Comparison
σ(n)−n = 28 vs n = 28
Classification
= Perfect
How it works
A perfect number equals the sum of its proper divisors (all divisors except itself). The first four perfect numbers are 6, 28, 496 and 8128. All known even perfect numbers have the form 2^(p−1) × (2^p − 1) where 2^p − 1 is a Mersenne prime. Whether odd perfect numbers exist is an open problem.
Formula
Perfect number definition
n is perfect ⟺ σ(n) − n = n ⟺ σ(n) = 2n
- n
- Number to test
- σ(n)
- Sum of all divisors of n
Frequently Asked Questions
What are the first few perfect numbers?
6 (1+2+3), 28 (1+2+4+7+14), 496, and 8128. The fifth is 33,550,336.
What is an abundant number?
A number whose proper divisors sum to more than itself. For example, 12 is abundant because 1+2+3+4+6 = 16 > 12. A number with divisor sum less than itself is called deficient.