Perfect number
From Number
This article defines a property that can be evaluated for a natural number, i.e., every natural number either satisfies the property or does not satisfy the property.
View a complete list of properties of natural numbers
Contents
Definition
A natural number is termed a perfect number if
, where
denotes the divisor sum function: the sum of all the positive divisors of
. In particular,
equals the sum of all its proper positive divisors.
Relation with other properties
Weaker properties
Variations
- Almost perfect number: This requires
.
- Quasiperfect number: This requires
.
Opposite properties
- Abundant number: This requires
.
- Deficient number: This requires
.
Facts
- If
(the
Mersenne number) is a prime number (and hence, a Mersenne prime), then
is a perfect number.
- Every even perfect number arises in the above fashion.
- The existence of odd perfect numbers is an open problem.