Editing Semiperfect number

{{Short description|Number equal to the sum of some of its divisors}}
{{Infobox integer sequence
| image = Perfect number Cuisenaire rods 6 exact.svg
| image_size = 250px
| caption = <small>Demonstration, with [[Cuisenaire rods]], of the perfection of the number 6.</small>
| number = [[infinity]]
| first_terms = [[6 (number)|6]], [[12 (number)|12]], [[18 (number)|18]], [[20 (number)|20]], [[24 (number)|24]], [[28 (number)|28]], [[30 (number)|30]]
| OEIS = A005835
| OEIS_name = Pseudoperfect (or semiperfect) numbers
}}
In [[number theory]], a '''semiperfect number''' or '''pseudoperfect number''' is a [[natural number]] ''n'' that is equal to the sum of all or some of its [[proper divisor]]s. A semiperfect number that is equal to the sum of all its proper divisors is a [[perfect number]].

The first few semiperfect numbers are: [[6 (number)|6]], [[12 (number)|12]], [[18 (number)|18]], [[20 (number)|20]], [[24 (number)|24]], [[28 (number)|28]], [[30 (number)|30]], [[36 (number)|36]], [[40 (number)|40]], ... {{OEIS|id=A005835}}

== Properties ==
* Every [[multiple (mathematics)|multiple]] of a semiperfect number is semiperfect.{{sfnp|Zachariou|Zachariou|1972}}  A semiperfect number that is not [[divisible]] by any smaller semiperfect number is called ''primitive''.
* Every number of the form 2<sup>''m''</sup>''p'' for a natural number ''m'' and an [[parity (mathematics)|odd]] [[prime number]] ''p'' such that ''p'' < 2<sup>''m''+1</sup> is also semiperfect. 
** In particular, every number of the form 2<sup>''m''</sup>(2<sup>''m''+1</sup>&nbsp;−&nbsp;1) is semiperfect, and indeed perfect if 2<sup>''m''+1</sup>&nbsp;−&nbsp;1 is a [[Mersenne prime]].
* The smallest odd semiperfect number is [[945 (number)|945]].{{sfnp|Friedman|1993}}
* A semiperfect number is necessarily either perfect or [[abundant number|abundant]]. An abundant number that is not semiperfect is called a [[weird number]]. 
* With the exception of 2, all [[primary pseudoperfect number]]s are semiperfect. 
* Every [[practical number]] that is not a [[power of two]] is semiperfect.
* The [[natural density]] of the [[set (mathematics)|set]] of semiperfect numbers exists.{{sfnp|Guy|2004|p=75}}

==Primitive semiperfect numbers==
A '''primitive semiperfect number''' (also called a ''primitive pseudoperfect number'', ''irreducible semiperfect number'' or ''irreducible pseudoperfect number'') is a semiperfect number that has no semiperfect proper divisor.{{sfnp|Guy|2004|p=75}}

The first few primitive semiperfect numbers are [[6 (number)|6]], [[20 (number)|20]], [[28 (number)|28]], [[88 (number)|88]], [[104 (number)|104]], [[272 (number)|272]], [[304 (number)|304]], [[350 (number)|350]], ... {{OEIS|A006036}}

There are infinitely many such numbers.  All numbers of the form 2<sup>''m''</sup>''p'', with ''p'' a prime between 2<sup>''m''</sup> and 2<sup>''m''+1</sup>, are primitive semiperfect, but this is not the only form: for example, 770.{{sfnp|Zachariou|Zachariou|1972}}{{sfnp|Guy|2004|p=75}}  There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of [[Paul Erdős]].{{sfnp|Guy|2004|p=75}} There are also infinitely many primitive semiperfect numbers that are not [[harmonic divisor number]]s.{{sfnp|Zachariou|Zachariou|1972}}

Every semiperfect number is a multiple of a primitive semiperfect number.

==See also==
* [[Hemiperfect number]]
* [[Erdős–Nicolas number]]

==Notes==
{{reflist}}

== References ==
* {{cite journal
 |title       = Sums of divisors and Egyptian fractions
 |last        = Friedman
 |first       = Charles N.
 |journal     = [[Journal of Number Theory]]
 |year        = 1993
 |volume      = 44
 |pages       = 328–339
 |mr          = 1233293
 |zbl         = 0781.11015
 |doi         = 10.1006/jnth.1993.1057
 |issue       = 3
 |doi-access= free
 }}
* {{cite book|last=Guy | first=Richard K. |author-link=Richard K. Guy|title=Unsolved Problems in Number Theory|publisher=[[Springer-Verlag]]|date=2004|isbn=0-387-20860-7|oclc=54611248 | zbl=1058.11001}} Section B2.
* {{ cite journal | last=Sierpiński | first=Wacław | author-link=Wacław Sierpiński | title=Sur les nombres pseudoparfaits | language=fr | journal=Mat. Vesn. |series=Nouvelle Série |volume=2 |issue=17 | pages=212–213 | year=1965 | zbl=0161.04402 | mr=199147 }}
* {{cite journal | zbl=0266.10012 | mr=360455 | last1=Zachariou | first1=Andreas | last2=Zachariou | first2=Eleni | title=Perfect, semiperfect and Ore numbers | journal=Bull. Soc. Math. Grèce |series=Nouvelle Série | volume=13 | pages=12–22 | year=1972 }}

== External links ==
* {{MathWorld |urlname=PseudoperfectNumber |title=Pseudoperfect Number}}
* {{MathWorld |urlname=PrimitivePseudoperfectNumber |title=Primitive Pseudoperfect Number}}

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

[[Category:Integer sequences]]
[[Category:Perfect numbers]]

[[de:Vollkommene Zahl#Pseudovollkommene Zahlen]]