Quasiperfect number

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far.

The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).

Theorems
If a quasiperfect number exists, it must be an odd square number greater than 1035 and have at least seven distinct prime factors.

Related
For a perfect number n the sum of all its divisors is equal to 2n.

For an almost perfect number n the sum of all its divisors is equal to 2n - 1.

Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.