Semiperfect number explained

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, ...

Properties

• Every multiple of a semiperfect number is semiperfect.^[1] A semiperfect number that is 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 indeed perfect if 2^m+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.^[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 6, 20, 28, 88, 104, 272, 304, 350, ...

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 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 Erdős:^[2] there are also infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.^[1]

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

See also

• Hemiperfect number
• Erdős–Nicolas number

References

• Sums of divisors and Egyptian fractions . Friedman . Charles N. . . 1993 . 44 . 328–339 . 1233293 . 0781.11015 . 10.1006/jnth.1993.1057 . 3 . free .
• Book: Guy, Richard K. . Richard K. Guy. Unsolved Problems in Number Theory. Springer-Verlag. 2004. 0-387-20860-7. 54611248 . 1058.11001. Section B2.
• Sierpiński . Wacław . Wacław Sierpiński . Sur les nombres pseudoparfaits . fr . Mat. Vesn. . Nouvelle Série . 2 . 17 . 212–213 . 1965 . 0161.04402 . 199147 .
• 0266.10012 . 360455 . Zachariou . Andreas . Zachariou . Eleni . Perfect, semiperfect and Ore numbers . Bull. Soc. Math. Grèce . Nouvelle Série . 13 . 12–22 . 1972 .

Notes and References

1. Zachariou+Zachariou (1972)
2. Guy (2004) p. 75