Semiperfect number

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, ... (sequence A005835 in OEIS)

Properties

• Every multiple of a semiperfect number is semiperfect. A semiperfect number that is not divisible by any smaller semiperfect number is a primitive semiperfect number.
• 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 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.

References

• Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory 44 (3): 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

• Weisstein, Eric W., "Pseudoperfect Number" from MathWorld.