Semiperfect number

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, ... Template: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 2^mp for a natural number m and an odd prime number p such that p < 2^m+1 is also semiperfect.
  • In particular, every number of the form 2^m(2^m+1 − 1) is semiperfect, and is indeed perfect if 2^m+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, ... Template:OEIS

There are infinitely many such numbers. All numbers of the form 2^mp, with p a prime between 2^m and 2^m+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

• Hemiperfect number
• Erdős–Nicolas number

References

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External links

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