Definition:Perfect Number

Definition
A perfect number is a natural number equal to the sum of all its divisors except itself.

Formally stated, a perfect number $n$ is a natural number such that $\sigma \left({n}\right) = 2 n$, where $\sigma: \N \to \N$ is the sigma function.

It is an open question as to whether all perfect numbers are even. No odd perfect numbers have ever been found. It is known that no odd perfect number contains less than 100 digits.

An even perfect number is of the form $2^{n-1} \left({2^n - 1}\right)$ as proved here, where $2^n - 1$ is prime.

Examples
The first few perfect numbers are:

Also see

 * Definition:Abundance