### A000396 - Perfect numbers n: n is equal to the sum of the proper divisors of n

A000396 - Perfect numbers n: n is equal to the sum of the proper divisors of n.

6, 28, 496, 8128, 33550336,…

Euclid, in his textbook *Elements*, was the first to record the construction of perfect numbers. He and his fellow ancient Greeks knew the first four (although a(4) = 8128 wasn’t recorded until a couple hundred years after *Elements* was first written). Almost two thousand years later, in 1588, the Italian mathemetician Pietro Cataldi calculated through the seventh perfect number.

In the 17th century, Leonhard Euler proved that there was a relationship between Mersenne primes (primes of the form 2^{p}−1 where *p* is also prime) and *even* perfect numbers. In fact, all even perfect numbers can be calculated from the formula 2^{p−1}(2^{p}−1) assuming 2^{p}−1 is a Mersenne prime. (Euclid conjectured that this was true, but it took a millennium to prove.) There are 47 known Mersenne primes, so there are 47 known even perfect numbers, the largest of which has almost 26 million digits.

All known perfect numbers are even, but it is still unknown whether or not perfect numbers can be odd. All odd numbers up through about 10^{300} have been checked, and some characteristics of a hypothetical odd perfect number are known, but it is still an open question.