Divisor Function — Definition, Formula & Examples

The divisor function

\sigma_k(n)

adds up the

k

-th powers of every positive divisor of

n

. Two especially common cases are

\sigma_0(n)

, which counts the number of divisors, and

\sigma_1(n)

, which gives their sum.

For a positive integer $n$ and a non-negative integer (or more generally, complex number) $k$ , the divisor function is defined as $\sigma_k(n) = \sum_{d \mid n} d^k$ , where the sum ranges over all positive divisors $d$ of $n$ . The special case $\sigma_0(n)$ is often denoted $d(n)$ or $\tau(n)$ , and $\sigma_1(n)$ is often written simply as $\sigma(n)$ .

Key Formula

\sigma_k(n) = \sum_{d \mid n} d^{\,k}

Where:

$n$ = A positive integer whose divisors are being summed
$k$ = The exponent applied to each divisor (commonly 0 or 1)
$d \mid n$ = d ranges over all positive divisors of n

How It Works

To compute

\sigma_k(n)

, first list every positive integer that divides

n

evenly. Then raise each divisor to the

k

-th power and add the results. When

n

has a prime factorization

n = p_1^{a_1} p_2^{a_2} \cdots p_r^{a_r}

, the divisor function is multiplicative, meaning

\sigma_k(n) = \prod_{i=1}^{r} \sigma_k(p_i^{a_i})

. For a prime power, there is a closed form:

\sigma_k(p^a) = 1 + p^k + p^{2k} + \cdots + p^{ak}

Worked Example

Problem: Compute σ₀(12), σ₁(12), and σ₂(12).

List divisors: The positive divisors of 12 are 1, 2, 3, 4, 6, and 12.

Compute σ₀(12): Raise each divisor to the 0th power (each equals 1) and sum.

\sigma_0(12) = 1+1+1+1+1+1 = 6

Compute σ₁(12): Sum the divisors themselves.

\sigma_1(12) = 1+2+3+4+6+12 = 28

Compute σ₂(12): Sum the squares of each divisor.

\sigma_2(12) = 1+4+9+16+36+144 = 210

Answer: σ₀(12) = 6, σ₁(12) = 28, σ₂(12) = 210.

Visualization

Why It Matters

The divisor function appears throughout analytic and algebraic number theory. Classifying perfect numbers relies on

\sigma_1(n) = 2n

, a condition tied to open problems like the existence of odd perfect numbers. It also arises in the theory of modular forms, where identities involving

\sigma_k

connect to partition counting and the Ramanujan tau function.

Common Mistakes

Mistake: Confusing σ₀(n) (number of divisors) with σ₁(n) (sum of divisors) because both are called 'the divisor function.'

Correction: Always check the subscript k. σ₀ counts divisors, σ₁ sums them, and σ_k for higher k sums their k-th powers. When no subscript is written, σ(n) typically means σ₁(n).