Perfect Numbers are numbers which are equal to the sum of their proper divisors.
For example, the first few perfect numbers are 6, 28, 496, 8128, ... since
6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14,
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064
The first eight perfect numbers are summarized in the following table:
| n | Number |
| 1 | 6 |
| 2 | 28 |
| 3 | 496 |
| 4 | 8128 |
| 5 | 33550336 |
| 6 | 8589869056 |
| 7 | 137438691328 |
| 8 | 2305843008139952128 |
In the Elements, Euclid showed that whenever the sum of doubles (1, 2, 4, 8, 16, 32, ...) is a prime number, then you can create a perfect number by multiplying the sum by the highest double that you added.
Here are some examples:
| Sum of Doubles | Multiply by the Highest Double to get Perfect Number |
| 1 + 2 = 3 | 3 is prime, so multiply by the highest double, which is 2: 3 x 2 = 6 |
| 1 + 2 + 4 = 7 | 7 is prime, so multiply 7 by 4 to get 28 |
| 1 + 2 + 4 + 8 =15 | 15 is not prime, so no perfect number here |
| 1 + 2 + 4 + 8 + 16 = 31 | 31 is prime, so multiply 31 x 16 = 496 |
| 1 + 2 + 4 + 8 + 16 + 32 = 63 | 63 is not prime, so no perfect number here |
| 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 | 127 is prime, so multiply 127 x 64 = 8,128 |
from Here's Looking at Euclid by Alex Bellos