A perfect number is a natural number $n > 0$ which equals the sum of its proper divisors, formally:
\[n=\sum_{d\mid n} d - n\Longleftrightarrow n\text{ is perfect}.\]
Equivalently, if we allow the non-proper divisor $n\mid n$ to contribute to the sum of divisors, $n$ is perfect if and only if the sum of all divisors equals $2n$, formally: $$2n=\sum_{d\mid n}d\Longleftrightarrow n\text{ is perfect}.$$
Definitions: 1
Problems: 2 3
Proofs: 4
Propositions: 5 6