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

Numbers n whose sum of factors equals ${\displaystyle 2n+2}${\displaystyle 2n+2} are known to exist. They are of form ${\displaystyle 2^{n-1}\times (2^{n}-3)}${\displaystyle 2^{n-1}\times (2^{n}-3)} where ${\displaystyle 2^{n}-3}${\displaystyle 2^{n}-3} is a prime. The only exception known so far is ${\displaystyle 650=2\times 5^{2}\times 13}${\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 ${\displaystyle 2n-2}${\displaystyle 2n-2} are also known to exist. They are of form ${\displaystyle 2^{n-1}\times (2^{n}+1)}${\displaystyle 2^{n-1}\times (2^{n}+1)} where ${\displaystyle 2^{n}+1}${\displaystyle 2^{n}+1} is prime. No exceptions are found so far. Because of the five known Fermat primes, there are five 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.

• Brown, E.; Abbott, H.; Aull, C.; Suryanarayana, D. (1973). "Quasiperfect numbers" (PDF). Acta Arith. 22 (4): 439–447. doi:10.4064/aa-22-4-439-447 . MR 0316368.
• 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.

• Arithmetic dynamics
• Divisor function
• Integer sequences
• Unsolved problems in number theory

• Articles with short description
• Short description is different from Wikidata