Sum of divisors

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 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, abundant and deficient 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 $\sigma(p) = p + 1$.