Editing Almost perfect number

{{Short description|Numbers whose sum of divisors is twice the number minus 1}}
[[File:Deficient number Cuisenaire rods 8.png|thumb|Demonstration, with [[Cuisenaire rods]], that the number 8 is almost perfect, and [[deficient number|deficient]].]]

In [[mathematics]], an '''almost perfect number''' (sometimes also called '''slightly defective''' or '''least deficient''' '''number''') is a [[natural number]] ''n'' such that the sum of all [[divisor]]s of ''n'' (the [[sum-of-divisors function]] ''σ''(''n'')) is equal to 2''n''&nbsp;−&thinsp;1, the sum of all [[proper divisor]]s of ''n'', ''s''(''n'') = ''σ''(''n'') − ''n'', then being equal to ''n''&nbsp;−&thinsp;1. The only known almost perfect numbers are [[power of two|powers of 2]] with non-negative exponents {{OEIS|A000079}}. Therefore the only known [[parity (mathematics)|odd]] almost perfect number is 2<sup>0</sup> = 1, and the only known even almost perfect numbers are those of the form 2<sup>''k''</sup> for some positive [[integer]] ''k''; however, it has not been shown that all almost perfect numbers are of this form.  It is known that an odd almost perfect number greater than 1 would have at least six [[prime factor]]s.<ref name=Kis1978>{{ 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 | 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 }}</ref><ref name=Kis1981>{{cite journal | last=Kishore | first=Masao | title=On odd perfect, quasiperfect, and odd almost perfect numbers | journal=[[Mathematics of Computation]] | volume=36 | pages=583–586 | year=1981 | issue=154 | issn=0025-5718 | zbl=0472.10007 | doi=10.2307/2007662| jstor=2007662 | doi-access=free }}</ref>

If ''m'' is an odd almost perfect number then {{nowrap|''m''(2''m'' − 1)}} is a [[Descartes number]].<ref name=BGNS>{{cite book | last1=Banks | first1=William D. | last2=Güloğlu | first2=Ahmet M. | last3=Nevans | first3=C. Wesley | last4=Saidak | first4=Filip | chapter=Descartes numbers | pages=167–173  | editor1-last=De Koninck | editor1-first=Jean-Marie | editor1-link=Jean-Marie De Koninck | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006 | location=Providence, RI | publisher=[[American Mathematical Society]] | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1186.11004 }}</ref> Moreover if ''a'' and ''b'' are positive odd integers such that <math>b+3<a<\sqrt{m/2}</math> and such that {{nowrap|4''m'' − ''a''}} and {{nowrap|4''m'' + ''b''}} are both [[prime number|primes]], then {{nowrap|''m''(4''m'' − ''a'')(4''m'' + ''b'')}} would be an odd [[weird number]].<ref>
{{cite journal
  | last =Melfi
  | first =Giuseppe
  | author-link=Giuseppe Melfi
  | title =On the conditional infiniteness of primitive weird numbers
  | journal =[[Journal of Number Theory]]
  | volume =147
  | pages = 508–514
  | year =2015
  | doi= 10.1016/j.jnt.2014.07.024
 | doi-access =free
  }}
</ref>

== See also ==
*[[Perfect number]]
*[[Quasiperfect number]]

== References ==
{{reflist}}

==Further reading==
* {{cite book | author-link=Richard K. Guy | last=Guy | first=R. K. | chapter=Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers <!-- |at=B2 -->| title=[[Unsolved Problems in Number Theory (book)|Unsolved Problems in Number Theory]] | edition=2nd | location=New York | publisher=[[Springer-Verlag]] | pages=16, 45–53 | year=1994 }}
* {{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 | page=110 }}
* {{cite book | editor1-last=Sándor | editor1-first=Jozsef | editor2-last=Crstici | editor2-first=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | pages=37–38 | zbl=1079.11001 }}
* {{cite book | last=Singh | first=S. | title=Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem | url=https://archive.org/details/fermatsenigmaepi00sing_0 | url-access=registration | location=New York | publisher=Walker | page=[https://archive.org/details/fermatsenigmaepi00sing_0/page/13 13] | year=1997 | isbn=9780802713315 }}

== External links ==
* {{mathworld|urlname=AlmostPerfectNumber|title=Almost perfect number}}

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

[[Category:Arithmetic dynamics]]
[[Category:Divisor function]]
[[Category:Integer sequences]]

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