Perfect number

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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

Variations

Opposite properties

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.