In mathematics, 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, ... ;

every multiple of a semiperfect number is semiperfect, and every number of the form 2^mp for a natural number m and a prime number p such that p < 2^{m + 1} is also semiperfect. In particular, every number of the form 2^m-1(2^m-1) is semiperfect, and indeed perfect if 2^m-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 which 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.

A semiperfect number that is not divisible by any smaller semiperfect number is a primitive semiperfect number.

References

• Section B2.

External links

• MathWorld: Semiperfect number

eo:Duonperfekta nombro fr:Nombre semi-parfait it:Numero semi-perfetto he:???? ???? ?????? nl:Semiperfect getal ja:????? ru:??????????????? ????? fi:Puolitäydellinen luku zh:????

---

Source: Wikipedia | The above article is available under the GNU FDL. | Edit this article