Mersenne primes are prime numbers that are formed by raising 2 to a power k and subtracting 1; in equation form that would be m = (2^k) -1. Multiplying the Mersenne prime (2^k) -1 and 2^(k - 1), which may or may not be a prime, gives a perfect number.
There are no known odd perfect numbers. Every number up to 300 digits long has been checked, but so far no one has proven that an odd perfect number cannot exist, so the possibility must remain open.
Social numbers are very similar to perfect numbers, but they must come as a pair. A social number is equal to the sum of its partner's divisors. For example, 284 is equal to the sum of the divisors of 220, and 220 is equal to the sum of the divisors of 284. This makes 220 and 284 social numbers.