Semiperfect number

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, ... (sequence A005835 in OEIS);

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

• Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory 44: 328–339. doi:10.1006/jnth.1993.1057. MR1233293. http://dell5.ma.utexas.edu/users/friedman/divisors.ps. 
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7. OCLC 54611248.  Section B2.
• Sierpiński, Wacław (1965). "Sur les nombres pseudoparfaits" (in French). Mat. Vesn., N. Ser. 2 17: 212–213. 

External links

• MathWorld: Semiperfect number