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In number theory, a perfect number is a natural number that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself).
1 + 2 + 3 = 6, so 6 is a perfect number.
1 + 2 + 4 + 7 + 14 = 28, so 28 is a perfect number.

The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself.


For example, the first four perfect numbers are generated by the formula 2^(p−1) * (2^p − 1), with p a prime number, as follows:

for p = 2: 21(22 − 1) = 2 × 3 = 6
for p = 3: 22(23 − 1) = 4 × 7 = 28
for p = 5: 24(25 − 1) = 16 × 31 = 496
for p = 7: 26(27 − 1) = 64 × 127 = 8128.

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