# Divisor Function

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The **divisor function** is an arithmetic function that returns the number of distinct positive integer divisors of a positive integer.

Let $n$ be a positive integer. The

divisor function$\sigma_0(n)$ is defined as$\sigma_0(n)=(\text{the number of positive integer divisors of }n).$

The divisor function is sometimes denoted as $d(n).$ $_\square$

A more general form of this function is the **sum of positive divisors function**, which returns the sum of powers of the distinct positive divisors of a positive integer.

Let $n$ be a positive integer and let $x$ be a real or complex number. The

sum of positive divisors functionis defined as$\sigma_x(n)=\sum\limits_{d \mid n}{d^x}.\ _\square$

$\sigma_1(n)$ is the **sum of divisors function**, which returns the sum of all positive divisors of a number. This is sometimes denoted as $\sigma(n).$

**Cite as:**Divisor Function.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/divisor-function/