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 the OEIS) 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.

[2] The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... (sequence A006036 in the OEIS) There are infinitely many such numbers.

[1][2] There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of Paul Erdős:[2] there are also infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.

[1] Every semiperfect number is a multiple of a primitive semiperfect number.