Semiperfect number: Difference between revisions
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{{Short description|Number equal to the sum of some of its divisors}} | {{Short description|Number equal to the sum of all or some of its divisors}} | ||
{{Infobox integer sequence | {{Infobox integer sequence | ||
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| OEIS_name = Pseudoperfect (or semiperfect) numbers | | OEIS_name = Pseudoperfect (or semiperfect) numbers | ||
}} | }} | ||
In [[number theory]], a '''semiperfect number''' or '''pseudoperfect number''' is a [[natural number]] ''n'' | In [[number theory]], a '''semiperfect number''' or '''pseudoperfect number''' is a [[natural number]] ''n'' equal to the sum of all or some of its [[proper divisor]]s. A semiperfect number 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}} | 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 == | == Properties == | ||
* Every [[multiple (mathematics)|multiple]] of a semiperfect number is semiperfect.{{sfnp|Zachariou|Zachariou|1972}} A semiperfect number | * Every [[multiple (mathematics)|multiple]] of a semiperfect number is semiperfect.{{sfnp|Zachariou|Zachariou|1972}} A semiperfect number 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. | * 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> − 1) is semiperfect, and indeed perfect if 2<sup>''m''+1</sup> − 1 is a [[Mersenne prime]]. | ** In particular, every number of the form 2<sup>''m''</sup>(2<sup>''m''+1</sup> − 1) is semiperfect, and is indeed perfect if 2<sup>''m''+1</sup> − 1 is a [[Mersenne prime]]. | ||
* The smallest odd semiperfect number is [[945 (number)|945]]. | * The smallest odd semiperfect number is [[945 (number)|945]]. | ||
* A semiperfect number is necessarily either perfect or [[abundant number|abundant]]. An abundant number that is not semiperfect is called a [[weird number]]. | * A semiperfect number is necessarily either perfect or [[abundant number|abundant]]. An abundant number that is not semiperfect is called a [[weird number]]. | ||
* | * Except for 2, all [[primary pseudoperfect number]]s are semiperfect. | ||
* Every [[practical number]] that is not a [[power of two]] is 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 | * The [[natural density]] of the [[set (mathematics)|set]] of semiperfect numbers exists.{{sfnp|Guy|2004}} | ||
==Primitive semiperfect numbers== | ==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 | 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}} | ||
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}} | 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 | 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 not all primitive semiperfect numbers follow this form; for example, [[770 (number)|770]].{{sfnp|Zachariou|Zachariou|1972}}{{sfnp|Guy|2004}} There are infinitely many odd primitive semiperfect numbers, the smallest being [[945 (number)|945]]. There are 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. | Every semiperfect number is a multiple of a primitive semiperfect number. | ||
Latest revision as of 09:00, 4 September 2025
Template:Short description Template:Infobox integer sequence In number theory, a semiperfect number or pseudoperfect number is a natural number n equal to the sum of all or some of its proper divisors. A semiperfect number equal to the sum of all its proper divisors is a perfect number.
The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (sequence A005835 in the OEIS)
Properties
- Every multiple of a semiperfect number is semiperfect.Template:Sfnp A semiperfect number not divisible by any smaller semiperfect number is called primitive.
- Every number of the form 2mp for a natural number m and an odd prime number p such that p < 2m+1 is also semiperfect.
- In particular, every number of the form 2m(2m+1 − 1) is semiperfect, and is indeed perfect if 2m+1 − 1 is a Mersenne prime.
- The smallest odd semiperfect number is 945.
- A semiperfect number is necessarily either perfect or abundant. An abundant number that is not semiperfect is called a weird number.
- Except for 2, all primary pseudoperfect numbers are semiperfect.
- Every practical number that is not a power of two is semiperfect.
- The natural density of the set of semiperfect numbers exists.Template:Sfnp
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.Template:Sfnp
The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... (sequence A006036 in the OEIS)
There are infinitely many such numbers. All numbers of the form 2mp, with p a prime between 2m and 2m+1, are primitive semiperfect, but not all primitive semiperfect numbers follow this form; for example, 770.Template:SfnpTemplate:Sfnp There are infinitely many odd primitive semiperfect numbers, the smallest being 945. There are infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.Template:Sfnp
Every semiperfect number is a multiple of a primitive semiperfect number.
See also
Notes
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References
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External links
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Template:Divisor classes Template:Classes of natural numbers