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.
|Total no. of terms||infinity|
|First terms||6, 12, 18, 20, 24, 28, 30|
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.
There are infinitely many such numbers. All numbers of the form 2mp, with p a prime between 2m and 2m+1, are primitive semiperfect, but this is not the only form: for example, 770. There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of Paul Erdős: there are also infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.
Every semiperfect number is a multiple of a primitive semiperfect number.