Divisor function

Definition

Let $k$ be a natural number. The divisor function $\sigma_k$ is the arithmetic function given by $$\sigma_k(n) = \sum_{d \mid n} d^k,$$ where the sum runs over the positive divisors $d$ of $n$.

Remarks

The case $k = 0$ is the number of divisors, and the case $k = 1$ is the sum of divisors. For example, $\sigma_2(6) = 1 + 4 + 9 + 36 = 50$. The same formula defines $\sigma_k$ for each real or complex number $k$. The function $\sigma_k$ is multiplicative.

Explanations

L1 human

To compute $\sigma_k(n)$, list the divisors of $n$, raise each one to the power $k$, and add the results.