Quasiperfect number

In

${\displaystyle \sigma (n)}$) is equal to ${\displaystyle 2n+1}$. 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

Related

For a perfect number n the sum of all its divisors is equal to ${\displaystyle 2n}$. For an almost perfect number n the sum of all its divisors is equal to ${\displaystyle 2n-1}$.

Numbers n whose sum of factors equals ${\displaystyle 2n+2}$ are known to exist. They are of form ${\displaystyle 2^{n-1}\times (2^{n}-3)}$ where ${\displaystyle 2^{n}-3}$ is a prime. The only exception known so far is ${\displaystyle 650=2\times 5^{2}\times 13}$. They are 20, 104, 464, 650, 1952, 130304, 522752, ... (sequence A088831 in the OEIS). Numbers n whose sum of factors equals ${\displaystyle 2n-2}$ are also known to exist. They are of form ${\displaystyle 2^{n-1}\times (2^{n}+1)}$ where ${\displaystyle 2^{n}+1}$ is prime. No exceptions are found so far. Because of the five known

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

References

• Brown, E.; Abbott, H.; Aull, C.; Suryanarayana, D. (1973). "Quasiperfect numbers" (PDF). Acta Arith. 22 (4): 439–447.
  MR 0316368
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• Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2+10⁻¹²" (PDF).
  Zbl 0376.10005
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• Cohen, Graeme L. (1980). "On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers". J. Austral. Math. Soc. Ser. A. 29 (3): 369–384.
  Zbl 0425.10005
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• James J. Tattersall (1999). Elementary number theory in nine chapters.
  Zbl 0958.11001
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• ISBN 0-387-20860-7
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• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht:
  Zbl 1151.11300
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