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 functionis 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