Perfect number: Difference between revisions
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A [[natural number]] <math>n</math> is termed a '''perfect number''' if <math>\sigma(n) = 2n</math>, where <math>\sigma</math> denotes the [[defining ingredient::divisor sum function]]: the sum of all the positive divisors of <math>n</math>. In particular, <math>n</math> equals the sum of all its ''proper'' positive divisors. | A [[natural number]] <math>n</math> is termed a '''perfect number''' if <math>\sigma(n) = 2n</math>, where <math>\sigma</math> denotes the [[defining ingredient::divisor sum function]]: the sum of all the positive divisors of <math>n</math>. In particular, <math>n</math> equals the sum of all its ''proper'' positive divisors. | ||
==Relation with other properties== | |||
===Opposite properties=== | |||
* [[Abundant number]] | |||
* [[Deficient number]] | |||
===Weaker properties=== | |||
* [[Stronger than::Pseudoperfect number]] | |||
==Facts== | ==Facts== | ||
Revision as of 23:18, 21 March 2009
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
Opposite properties
Weaker 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.
