Superperfect number

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

σ 2 ( n ) = σ ( σ ( n ) ) = 2 n ,

where σ is the divisor summatory 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, 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.

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 7×10²⁴.