---
tag: 009QCP5T3Z
kind: definition
name: Sum of divisors
aliases: [sum-of-divisors-function]
msc: [11A25]
defines: 06TK51263Q
uses: [0YEW47VJ0J]
sources:
  - url: https://en.wikipedia.org/wiki/Divisor_function
    locator: Definition
---

## Definition

For a positive integer $n$, the sum of divisors of $n$ is
$$\sigma(n) = \sigma_1(n) = \sum_{d \mid n} d,$$
where $\sigma_1$ is the [[divisor function|0YEW47VJ0J]] with $k = 1$.

## Remarks

If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$, then $\sigma(n) = \prod_{i=1}^{r} \frac{p_i^{e_i + 1} - 1}{p_i - 1}$.
For example, $\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$.
The sum of the proper divisors is $\sigma(n) - n$. It defines [[perfect|0SB7Q8SYSF]], [[abundant|0VS0MP1P5H]] and [[deficient|0G01TS1RRP]] numbers.

## Explanations

### L1 human

The function $\sigma(n)$ adds all the divisors of $n$, including $1$ and $n$.
For a prime $p$, the only divisors are $1$ and $p$, so $\sigma(p) = p + 1$.