Editing Semiperfect number (section)

==Primitive semiperfect numbers==
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.{{sfnp|Guy|2004|p=75}}

The first few primitive semiperfect numbers are [[6 (number)|6]], [[20 (number)|20]], [[28 (number)|28]], [[88 (number)|88]], [[104 (number)|104]], [[272 (number)|272]], [[304 (number)|304]], [[350 (number)|350]], ... {{OEIS|A006036}}

There are infinitely many such numbers.  All numbers of the form 2<sup>''m''</sup>''p'', with ''p'' a prime between 2<sup>''m''</sup> and 2<sup>''m''+1</sup>, are primitive semiperfect, but this is not the only form: for example, 770.{{sfnp|Zachariou|Zachariou|1972}}{{sfnp|Guy|2004|p=75}}  There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of [[Paul Erdős]].{{sfnp|Guy|2004|p=75}} There are also infinitely many primitive semiperfect numbers that are not [[harmonic divisor number]]s.{{sfnp|Zachariou|Zachariou|1972}}

Every semiperfect number is a multiple of a primitive semiperfect number.