Superperfect number


In mathematics, a superperfect number is a positive integer n that satisfies
where σ is the divisor summatory function. Superperfect numbers are a generalization of perfect numbers. The term was coined by D. Suryanarayana.
The first few superperfect numbers are :
To illustrate: it can be seen that 16 is a superperfect number as σ = 1 + 2 + 4 + 8 + 16 = 31, and σ = 1 + 31 = 32, thus σ = 32 = 2 × 16.
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.
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 is divisible by at least three distinct primes. There are no odd superperfect numbers below 7.

Generalizations

Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy
corresponding to m=1 and 2 respectively. For m ≥ 3 there are no even m-superperfect numbers.
The m-superperfect numbers are in turn examples of -perfect numbers which satisfy
With this notation, perfect numbers are -perfect, multiperfect numbers are -perfect, superperfect numbers are -perfect and m-superperfect numbers are -perfect. Examples of classes of -perfect numbers are: