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 $n$ is termed a perfect number if $\sigma (n)=2n$, where $\sigma$ denotes the divisor sum function: the sum of all the positive divisors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . In particular, $n$ equals the sum of all its proper positive divisors.

Relation with other properties

Weaker properties

• Pseudoperfect number

Variations

• Almost perfect number: This requires $\sigma (n)=2n-1$.
• Quasiperfect number: This requires $\sigma (n)=2n+1$.

Opposite properties

• Abundant number: This requires $\sigma (n)>2n$.
• Deficient number: This requires $\sigma (n)<2n$.

Facts

• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_n = 2^n - 1} (the $n^{th}$ Mersenne number) is a prime number (and hence, a Mersenne prime), then $2^{n-1}(2^{n}-1)$ is a perfect number.
• Every even perfect number arises in the above fashion.
• The existence of odd perfect numbers is an open problem.