Surf Wiki
Save to docs
general/integer-sequences

From Surf Wiki (app.surf) — the open knowledge base

Semiperfect number

Number equal to the sum of all or some of its divisors

Number equal to the sum of all or some of its divisors

FieldValue
imagePerfect number Cuisenaire rods 6 exact.svg
image_size250px
captionDemonstration, with Cuisenaire rods, of the perfection of the number 6.
numberinfinity
first_terms6, 12, 18, 20, 24, 28, 30
OEISA005835
OEIS_namePseudoperfect (or semiperfect) numbers

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, ...

Properties

  • Every multiple of a semiperfect number is semiperfect. A semiperfect number not divisible by any smaller semiperfect number is called primitive.
  • Every number of the form 2m**p for a natural number m and an odd prime number p such that p m+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.

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, ...

There are infinitely many such numbers. All numbers of the form 2m**p, with p a prime between 2m and 2m+1, are primitive semiperfect, but not all primitive semiperfect numbers follow this form; for example, 770. 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.

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

Notes

References

  • {{cite journal |doi-access= free
  • Section B2.
Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

integer-sequences perfect-numbers
Want to explore this topic further?

Ask Mako anything about Semiperfect number — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report