"Perfect Numbers" Defined

The first eight perfect numbers are
6
28
496
8,128
33,550,336
8,589,869,056
137,438,691,328
2,305,843,008,139,952,128

As of 2013, there are 48 known perfect numbers, all of them even integers. All of these known perfect numbers end in "6" or "28." See
 

http://en.wikipedia.org/wiki/List_of_perfect_numbers

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Perfect Numbers
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ₁(n) = 2n.

This definition is ancient, appearing as early as Euclid’s Elements (VII.22) where it is called τέλειος αριθμός
(perfect, ideal, or complete number). Much later, Euler proved that all even perfect numbers are of this form.
It is not known whether there are any odd perfect numbers.

Examples
The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128

Discovery
These first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus had noted 8128 as early as 100 AD. In a manuscript written between 1456 and 1461, an unknown mathematician recorded the earliest reference to a fifth perfect number, with 33,550,336 being correctly identified for the first time. In 1588, the Italian mathematician Pietro Cataldi identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers.

Odd Perfect Numbers
It is unknown whether there are any odd perfect numbers, though various results have been obtained. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. All perfect numbers are also Ore’s harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1.

Related Concepts
The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.

By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence.
All perfect numbers are also Granville numbers.

This information was simplified from:
  http://en.wikipedia.org/wiki/Perfect_numbers