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In mathematics, a quasiperfect number is a theoretical 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').
The quasiperfect numbers are the abundant numbers of minimal abundance 1.
No quasiperfect numbers have been found so far, but if a quasiperfect number exists, it must be an odd square number greater than 1035 and have at least seven distinct prime factors.
Numbers do exist where the sum of all the divisors σ(n) is equal to 2n + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... (sequence A088831 in the OEIS). Many of these numbers are of the form 2n−1(2n − 3) where 2n − 3 is prime (instead of 2n − 1 with perfect numbers)