Superperfect Number

Superperfect Number

In mathematics a superperfect number is a positive integer n that satisfies

where σ is the divisor function. Superperfect numbers are a generalization of perfect numbers. The term was coined by Suryanarayana (1969).

The first few superperfect numbers are

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

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(n) is divisible by at least three distinct primes. There are no odd superperfect numbers below 7x1024.