Divisor Function

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 function is 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 $\sigma(n).$