In mathematics, a quasiperfect number is a natural number for which the sum of all its divisors (the sum-of-divisors function ) is equal to . Equivalently, is the sum of its non-trivial divisors (that is, its divisors excluding 1 and ). No quasiperfect numbers have been found so far.
The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).
If a quasiperfect number exists, it must be an odd square number greater than 10<sup>35</sup> and have at least seven distinct prime factors.
For a perfect number the sum of all its divisors is equal to . For an almost perfect number the sum of all its divisors is equal to .
Numbers whose sum of factors equals are known to exist. They are of form where is a prime. The only exception known so far is . They are 20, 104, 464, 650, 1952, 130304, 522752, ... . Numbers whose sum of factors equals are also known to exist. They are of form where is prime. No exceptions are found so far. Because of the five known Fermat primes, there are five such numbers known: 3, 10, 136, 32896 and 2147516416
Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.