Quasiperfect number

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In mathematics, 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).

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.[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)

Notes[edit]

  1. ^ Hagis, Peter; Cohen, Graeme L. (1982). "Some results concerning quasiperfect numbers". J. Austral. Math. Soc. Ser. A. 33 (2): 275–286. MR 0668448. doi:10.1017/S1446788700018401. 

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