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 = Demonstration, with Cuisenaire rods, of the perfection of the number 6.
| number = infinity
| first_terms = 6, 12, 18, 20, 24, 28, 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 divisors. 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, 12, 18, 20, 24, 28, 30, 36, 40, ... {{OEIS|id=A005835}}
Properties
- Every multiple of a semiperfect number is semiperfect.Zachariou+Zachariou (1972) A semiperfect number that is 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 indeed perfect if 2m+1 − 1 is a Mersenne prime.
- The smallest odd semiperfect number is 945 (see, e.g., Friedman 1993).
- A semiperfect number is necessarily either perfect or abundant. An abundant number that is not semiperfect is called a weird number.
- With the exception of 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.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.
The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... {{OEIS|A006036}}
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 this is not the only form: for example, 770. There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of Paul Erdős: there are also infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.
Every semiperfect number is a multiple of a primitive semiperfect number.
See also
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=PrimitiveSemiperfectNumber |title=Primitive semiperfect number}}
{{Divisor classes}}
{{Classes of natural numbers}}