查看“︁殆完全數”︁的源代码

{{NoteTA|G1=Math}}
[[File:Deficient number Cuisenaire rods 8.png|thumb|用[[古氏积木]]說明8是殆完全數，也是[[亏数]]]]

{{unsolved|數學|不是[[2的冪]]的殆完全數存在嗎？}}

'''殆完全數'''（{{lang|en|'''almost perfect number'''}}）是一種特別的[[自然數]]，它所有的[[真因數和|真因數（即除了自身以外的因數）的和]]，恰好等於它本身減一。若用[[除數函數]]（其真因數的和及其本身）來表示，若一自然數''n''的[[除數函數]]''σ''(''n'')等於2''n'' - 1，該自然數即為殆完全數。殆完全數是一種[[虧數]]。虧度（σ(n) − 2n）為-1。

例如4的除數函數為2+1=3，比4小1，因此4是殆完全數。

目前已知的殆完全數為2的非負次幂{{OEIS|id=A000079}}，因此唯一已知奇數的殆完全數為2<sup>0</sup> = 1，但尚未證明除了2的非負次幂以外，是否存在其他型式的殆完全數。可以證明若存在大於1的奇數殆完全數，至少會有六個[[質因數]]<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>。

若''m''是奇數殆完全數，則{{nowrap|''m''(2''m'' − 1)}}會是[[笛卡爾數]]<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  | editor2-last=Granville | editor2-first=Andrew  | 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=[[美國數學學會|American Mathematical Society]] | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1186.11004 }}</ref>，而且，若''a''和''b''滿足<math>b+3<a<\sqrt{m/2}</math>，且{{nowrap|4''m'' − ''a''}} and {{nowrap|4''m'' + ''b''}}都是[[质数]]，則{{nowrap|''m''(4''m'' − ''a'')(4''m'' + ''b'')}}會是奇數的[[奇異數 (數論)|奇異數]]<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>。

==参见==
*[[完全數]]
*[[过剩数]]
*[[亏数]]
*[[準完全數]]

==參考資料==
* Richard K. Guy|Guy, R. K., ''Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers.'' §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 16 and 45-53, 1994.
* Singh, S., ''Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.'' New York: Walker, p. 13, 1997.
{{reflist}}
== 外部連結 ==
* {{mathworld|urlname=AlmostPerfectNumber|title=Almost perfect number}}

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