Quasiperfect number

If a quasiperfect number exists, it must be an odd square number greater than 10³⁵ and have at least seven distinct prime factors.[1]

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). In addition, numbers exist where the sum of all the divisors σ(n) is equal to 2n − 1, such as the powers of 2.

Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.

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