What is a Perfect Number?

A number is called perfect if it is equal to the sum of its divisors.
Six is perfect: 6 = 1 + 2 + 3.
28 is perfect: 28 = 1 + 2 + 4 + 7 + 14.

N is an even perfect number if and only if

          N = 2^(q-1) * (2^q - 1)   and   2^q - 1 is prime

It should also be noted that for 2^q - 1 to be prime q must be prime.
So when we search for even perfect numbers, we search on q equal to the primes.

The numbers M(q) = 2^q - 1 (with q prime) are called Mersenne numbers.
If M(q) = is prime then it is called a Mersenne prime.
If q makes a Mersenne prime then P(q) = 2^(q-1) * (2^q - 1) is a Perfect number.

The first few consecutive perfect numbers are -

P( 2) = 2 * 3 = 6
P( 3) = 4 * 7 = 28
P( 5) = 16 * 31 = 496
P( 7) = 64 * 127 = 8128
P(13) = 4096 * 8191 = 33,550,336
P(17) = 65,536 * 131,071 = 8,589,869,056
P(19) = 262,144 * 524,287 = 137,438,691,328
P(31) = 1,073,741,824 * 2,147,483,647 = 2,305,843,008,139,952,128
P(61) = 1,152,921,504,606,846,976 * 2,305,843,009,213,693,951 =
        2,658,455,991,569,831,744,654,692,615,953,842,176

P(61) can be computed with the program XICalc by:

Command: p=PEq(61)

p = 26,58455,99156,98317,44654,69261,59538,42176 (37 Digits)

See q for all known perfect numbers

It was pointed out to me by V. V. Raman that 

	"Every Perfect Number is 1 (mod 9).  V. V. Raman (1985)."

This is true for all even perfect numbers except 6, but actually

	2^(q-1) * (2^q - 1) = 1 (mod 9) iff q is an odd positive integer.

This is easy to prove. For q = 1, 3, 5, ...
2^(q-1) mod 9 = 1, 4, 7, 1, 4, 7, ... and repeats the 1, 4, 7,
2^q - 1 mod 9 = 1, 7, 4, 1, 7, 4, ... and repeats the 1, 7, 4,
so their product mod 9 = 1, 1, 1, 1, 1, 1 and repeats the 1.