In number theory, a superperfect number is a positive integer n that satisfies

${\displaystyle \sigma ^{2}(n)=\sigma (\sigma (n))=2n\,,}$

where σ is the divisor summatory function. Superperfect numbers are not a generalization of perfect numbers but have a common generalization. The term was coined by D. Suryanarayana (1969).^[1]

The first few superperfect numbers are :

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS).

To illustrate: it can be seen that 16 is a superperfect number as σ(16) = 1 + 2 + 4 + 8 + 16 = 31, and σ(31) = 1 + 31 = 32, thus σ(σ(16)) = 32 = 2 × 16.

If n is an even superperfect number, then n must be a power of 2, 2^k, such that 2^k+1 − 1 is a Mersenne prime.^[1][2]

It is not known whether there are any odd superperfect numbers. 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.^[2] There are no odd superperfect numbers below 7×10²⁴.^[1]

Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy

${\displaystyle \sigma ^{m}(n)=2n,}$

corresponding to m=1 and 2 respectively. For m ≥ 3 there are no even m-superperfect numbers.^[1]

The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy^[3]

${\displaystyle \sigma ^{m}(n)=kn\,.}$

With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect.^[4] Examples of classes of (m,k)-perfect numbers are: