Perfect, abundant or deficient number checker
A perfect number equals the sum of its proper divisors, like 28 = 1 + 2 + 4 + 7 + 14. Enter a number to see its class and divisor sum.
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How to use it
- Enter number.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
s(n) = σ(n) - n, the sum of proper divisors. Perfect if s(n) = n, abundant if s(n) > n, deficient if s(n) < n.
Example
Inputs: Number = 28. Result: Class: Perfect; Sum of proper divisors: 28; Difference from the number: 0.
Limits and caveats
- Accepts whole numbers from 1 to 10^12.
- The known perfect numbers in this range are 6, 28, 496, 8128, 33550336 and 8589869056.
Questions about this tool
What is the smallest perfect number?
6, since its proper divisors 1, 2 and 3 add up to 6.
What is an abundant number?
One whose proper divisors add up to more than itself, like 12 with a sum of 16. The smallest odd abundant number is 945.
Are there odd perfect numbers?
None has ever been found, and no one has proved that none exist.