Editing Perfect number (section)

== Related concepts ==
{{Euler_diagram_numbers_with_many_divisors.svg}}
The sum of [[proper divisor]]s gives various other kinds of numbers. Numbers where the sum is less than the number itself are called [[deficient number|deficient]], and where it is greater than the number, [[abundant number|abundant]]. These terms, together with ''perfect'' itself, come from Greek [[numerology]]. A pair of numbers which are the sum of each other's proper divisors are called [[amicable number|amicable]], and larger cycles of numbers are called [[sociable number|sociable]]. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a [[practical number]].

By definition, a perfect number is a [[fixed point (mathematics)|fixed point]] of the [[restricted divisor function]] {{nowrap|1=''s''(''n'') = ''σ''(''n'') − ''n''}}, and the [[aliquot sequence]] associated with a perfect number is a constant sequence.  All perfect numbers are also <math>\mathcal{S}</math>-perfect numbers, or [[Granville number]]s.

A [[semiperfect number]] is a natural number 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.  Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called [[weird number]]s.