Editing Quasiperfect number

{{Short description|Numbers whose sum of divisors is twice the number plus 1}}
In [[mathematics]], a '''quasiperfect number''' is a [[natural number]] ''n'' for which the sum of all its [[divisor]]s (the [[sum-of-divisors function]] ''σ''(''n'')) is equal to 2''n'' + 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 number]]s of minimal abundance (which is 1).

== Theorems ==
If a quasiperfect number exists, it must be an [[parity (mathematics)|odd]] [[square number]] greater than 10<sup>35</sup> and have at least seven distinct [[prime factor]]s.<ref>{{cite journal|last1=Hagis|first1=Peter |last2=Cohen|first2=Graeme L.|title=Some results concerning quasiperfect numbers|journal=J. Austral. Math. Soc. Ser. A|volume=33|year=1982|pages=275–286|doi=10.1017/S1446788700018401|issue=2|mr=0668448|doi-access=free}}</ref>

== Related ==
For a [[perfect number]] ''n'' the sum of all its divisors is equal to 2''n''. For an [[almost perfect number]] ''n'' the sum of all its divisors is equal to 2''n'' - 1.

Numbers n exist whose sum of factors = 2n + 2. They are of form 2^(n - 1) * (2^n - 3) where 2^n - 3 is a prime. The only exception known till yet is 650 = 2 * 5^2 * 13. They are 20, 104, 464, 650, 1952, 130304, 522752, etc. (OEIS [[oeis:A088831|A088831]]) Numbers n exist whose sum of factors = 2n - 2. They are of form 2^(n - 1) * (2^n + 1) where 2^n + 1 is prime. No exceptions are found till yet. Because of 5 known Fermat Primes, there are 5 known such numbers: 3, 10, 136, 32896 and 2147516416. (OEIS [[oeis:A191363|A191363]])

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

==Notes==
<references/>

== References ==
* {{cite journal|first1=E. |last1=Brown
|first2=H. |last2=Abbott
|first3=C. |last3=Aull
|first4=D. |last4=Suryanarayana
|title=Quasiperfect numbers
|journal=Acta Arith.
|year=1973
|volume=22
|issue=4
|pages=439–447
|mr=0316368
|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa22/aa2245.pdf |doi=10.4064/aa-22-4-439-447
|doi-access=free
}}
* {{cite journal 
| last=Kishore 
| first=Masao 
| title=Odd integers ''N'' with five distinct prime factors for which 2−10<sup>−12</sup> < σ(''N'')/''N'' < 2+10<sup>−12</sup> 
| journal=[[Mathematics of Computation]] 
| volume=32 
| issue=141 
| pages=303–309 
| year=1978 
| issn=0025-5718 
| zbl=0376.10005 
| mr=0485658
| url=https://www.ams.org/journals/mcom/1978-32-141/S0025-5718-1978-0485658-X/S0025-5718-1978-0485658-X.pdf 
| doi=10.2307/2006281
| jstor=2006281 
}}
* {{cite journal|first1=Graeme L.
|last1=Cohen|title= On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers 
|year=1980
|journal=J. Austral. Math. Soc. Ser. A 
|volume=29 
|issue=3|pages=369–384 
|doi=10.1017/S1446788700021376 
| mr=0569525 
| zbl=0425.10005 
|s2cid=120459203| issn=0263-6115
 }}
* {{cite book 
| author=James J. Tattersall 
| title=Elementary number theory in nine chapters 
| url=https://archive.org/details/elementarynumber00tatt_470
| url-access=limited
| publisher=[[Cambridge University Press]] 
| isbn=0-521-58531-7 
| year=1999 
| pages=[https://archive.org/details/elementarynumber00tatt_470/page/n156 147]
| zbl=0958.11001 }}
* {{cite book
 | last = Guy
 | first = Richard
 | author-link = Richard K. Guy
 | year = 2004
 | title = Unsolved Problems in Number Theory, third edition
 |page=74
 | publisher = [[Springer-Verlag]]
 | isbn=0-387-20860-7
}}
* {{cite book
 | editor1-last=Sándor 
| editor1-first=József 
| editor2-last=Mitrinović 
| editor2-first=Dragoslav S. 
| editor3-last=Crstici 
|editor3-first=Borislav 
| title=Handbook of number theory I 
| location=Dordrecht 
| publisher=[[Springer-Verlag]] 
| year=2006 
| isbn=1-4020-4215-9 
| zbl=1151.11300 
| pages=109–110
}}

{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Arithmetic dynamics]]
[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Unsolved problems in mathematics]]


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