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 2"n" + 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.
Numbers do exist where the sum of all the divisors "σ"("n") is equal to 2"n" + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... . 
Many of these numbers are of the form 2"n"−1(2"n" − 3) where 2"n" − 3 is prime (instead of 2"n" − 1 with perfect numbers). In addition, numbers exist where the sum of all the divisors "σ"("n") is equal to 2"n" − 1, such as the powers of 2.
Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.