Divisor function

Divisor function σ₀(n) up to n = 250

Sigma function σ₁(n) up to n = 250

Sum of the squares of divisors, σ₂(n), up to n = 250

Sum of cubes of divisors, σ₃(n) up to n = 250

In mathematics, and specifically 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). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.

A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.

Definition[]

The sum of positive divisors function σ_x(n), for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as

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

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^[1][2] (OEIS: A000005). When x is 1, the function is called the sigma function or sum-of-divisors function,^[1][3] 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.

Example[]

For example, σ₀(12) is the number of the divisors of 12:

${\displaystyle {\begin{aligned}\sigma _{0}(12)&=1^{0}+2^{0}+3^{0}+4^{0}+6^{0}+12^{0}\\&=1+1+1+1+1+1=6,\end{aligned}}}$

while σ₁(12) is the sum of all the divisors:

${\displaystyle {\begin{aligned}\sigma _{1}(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}+12^{1}\\&=1+2+3+4+6+12=28,\end{aligned}}}$

and the aliquot sum s(12) of proper divisors is:

${\displaystyle {\begin{aligned}s(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}\\&=1+2+3+4+6=16.\end{aligned}}}$

Table of values[]

The cases x = 2 to 5 are listed in OEIS: A001157 − OEIS: A001160, x = 6 to 24 are listed in OEIS: A013954 − OEIS: A013972.