Primitive semiperfect number

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

OEIS)

Properties

  • Every multiple of a semiperfect number is semiperfect.[1] A semiperfect number that is not divisible by any smaller semiperfect number is primitive.
  • Every number of the form 2mp for a natural number m and a prime number p such that p < 2m + 1 is also semiperfect.
    • In particular, every number of the form 2m − 1(2m − 1) is semiperfect, and indeed perfect if 2m − 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.[2]

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.[2]

The first few primitive semiperfect numbers are 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 this is not the only form: for example, 770.[1][2] There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of Paul Erdos:[2] there are also infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.[1]

See also

  • Hemiperfect number

Notes

References

  • Section B2.

External links

  • MathWorld.
  • MathWorld.


Template:Classes of natural numbersde:Vollkommene Zahl#Pseudovollkommene Zahlen

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