Semiperfect number

Properties

• 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 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 is semiperfect, and indeed perfect if 2^{m + 1} − 1 is a Mersenne prime.
• The smallest odd semiperfect number is 945.
• 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