Quasiperfect number


In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors. No quasiperfect numbers have been found so far.
The quasiperfect numbers are the abundant numbers of minimal abundance.

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