A number n such that σ^2(n) = σ(σ(n)) = 2n, where σ(n) is the divisor function is called a superperfect number. Even superperfect numbers are just 2^(p - 1), where M_p = 2^p - 1 is a Mersenne prime. If any odd superperfect numbers exist, they are square numbers and either n or σ(n) is divisible by at least three distinct primes. More generally, an m-superperfect (or (m, 2)-superperfect) number is a number for which σ^m(n) = 2n, and an (m, k)-perfect number is a number n for which σ^m(n) = k n.

divisor function | Mersenne number | perfect number