Perfect 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

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

• Pseudoperfect number

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.