Perfect Number Checker
Determine whether a number equals the sum of its proper divisors.
About Perfect Number Checker
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
Worked example
With the default values below, Perfect Number Checker returns Is Perfect? of Yes. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Numberminimum 1
- 28
Results
- Is Perfect?
- Yes
- Sum of Proper Divisors
- 28
- Classification
- Perfect
- Proper Divisors
- 1, 2, 4, 7, 14
Step by step
- 1Proper divisors
- 2Sum of proper divisors= 28
- 3Comparison
- 4Classification= Perfect
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.