How to Use the Divisor Calculator
Enter a positive integer n to instantly list all its divisors, compute the divisor count and sum σ(n), and classify n as perfect, deficient, or abundant. A perfect number equals the sum of its proper divisors (6, 28, 496...).
Number Classification by Divisor Sum
- Perfect: σ(n) = 2n (proper divisor sum equals n). Examples: 6, 28, 496
- Deficient: σ(n) < 2n. All primes are deficient
- Abundant: σ(n) > 2n. Example: 12 (1+2+3+4+6=16 > 12)
Frequently Asked Questions
Is there a quick formula for divisor count?
Factor n as p₁^a₁×p₂^a₂×... The divisor count is (a₁+1)(a₂+1)... Example: 12=2²×3¹ has (2+1)(1+1)=6 divisors.
How many perfect numbers are known?
As of 2025, 51 perfect numbers are known, all even. Whether any odd perfect number exists remains an open problem in mathematics.
What is the smallest abundant number?
12 is the smallest abundant number. Its proper divisors are 1, 2, 3, 4, 6, which sum to 16 — greater than 12 itself.