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Quasiperfect number

From Wikipedia, the free encyclopedia

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the sum-of-divisors 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).

Theorems

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

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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.

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

Notes

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

References

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