# Divisor Function

<https://en.wikipedia.org/wiki/Divisor_function>

In number theory, a **divisor function** is an arithmetic function related to the divisors of an integer.

When referred to as **the divisor function**, it counts the number of divisors of an integer (including 1 and the number itself).

**Divisor summatory function**is a sum over the divisor function.

The various studies of the behaviour of the divisor function are sometimes called **divisor problems**.

The sum of positive divisors function, $$σx(n)$$, for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of integer n. In sigma notation:

$${\displaystyle \sigma \_{x}(n)=\sum \_{d\mid n}d^{x},!,}$$

where $${\displaystyle {d\mid n}}$$ is shorthand for "d divides n".

The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ0(n), or the number-of-divisors function (OEIS: A000005).

When x is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ(n) is the same as σ1(n) (OEIS: A000203).

The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, OEIS: A001065), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.

## Euclid's GCD algorithm

Euclid's algorithm for determining the greatest common divisor (GCD) of two natural numbers.

To determine the GCD of two natural numbers:

* call the larger `x` and the smaller `y`
* let `m` be the remainder after dividing `x` by `y`
* if `m` is 0, `y` divides `x`, so `y` is the GCD of `x` and `y`
* otherwise, the GCD of `x` and `y` is equal to the GCD of `y` and `m`