Inmathematics, 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).
If a quasiperfect number exists, it must be an odd square number greater than 1035 and have at least seven distinct prime factors.[1]
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.
Numbers do exist where the sum of all the divisors is equal to 2n + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... (sequence A088831 in the OEIS). All numbers of the form 2n − 1(2n − 3) where 2n − 3 is prime belong to the sequence. As of 2024[update], the only known number of a different form in the sequence is 650 = 2 * 52 * 13.
Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.
Divisibility-based sets of integers
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Factorization forms |
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Constrained divisor sums |
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With many divisors |
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Aliquot sequence-related |
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Base-dependent |
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Other sets |
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Classes of natural numbers
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