Quasiperfect number

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the sum-of-divisors function ${\displaystyle \sigma (n)}$) 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 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 ${\displaystyle 2n}$. 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 5 known Fermat primes, there are 5 such numbers known: 3, 10, 136, 32896 and 2147516416 (sequence A191363 in the OEIS)

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

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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). Mathematics of Computation. 32 (141): 303–309. doi:10.2307/2006281. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005.
• Cohen, Graeme L. (1980). "On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers". J. Austral. Math. Soc. Ser. A. 29 (3): 369–384. doi:10.1017/S1446788700021376. ISSN 0263-6115. MR 0569525. S2CID 120459203. Zbl 0425.10005.
• James J. Tattersall (1999). Elementary number theory in nine chapters. Cambridge University Press. pp. 147. ISBN 0-521-58531-7. Zbl 0958.11001.
• Guy, Richard (2004). Unsolved Problems in Number Theory, third edition. Springer-Verlag. p. 74. ISBN 0-387-20860-7.
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 109–110. ISBN 1-4020-4215-9. Zbl 1151.11300.

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