In number theory, a superperfect number is a positive integer n that satisfies where σ is the sum-of-divisors function.
Superperfect numbers are not a generalization of perfect numbers but have a common generalization.
If n is an even superperfect number, then n must be a power of 2, 2k, such that 2k+1 − 1 is a Mersenne prime.
An odd superperfect number n would have to be a square number such that either n or σ(n) is divisible by at least three distinct primes.
[1] Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy corresponding to m = 1 and 2 respectively.