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
- Every multiple of a semiperfect number is semiperfect. 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 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.
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.
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.
- Zachariou+Zachariou (1972)
- Guy (2004) p. 75
- Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory. 44 (3): 328–339. doi:10.1006/jnth.1993.1057. MR 1233293. Zbl 0781.11015. Archived from the original on 2012-02-10.
- Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7. OCLC 54611248. Zbl 1058.11001. Section B2.
- Sierpiński, Wacław (1965). "Sur les nombres pseudoparfaits". Mat. Vesn., N. Ser. 2 (in French). 17: 212–213. MR 0199147. Zbl 0161.04402.
- Zachariou, Andreas; Zachariou, Eleni (1972). "Perfect, semiperfect and Ore numbers". Bull. Soc. Math. Grèce, n. Ser. 13: 12–22. MR 0360455. Zbl 0266.10012.