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 \(\sigma(n).\)