Number of divisors

Definition

For a positive integer $n$, the number of divisors of $n$ is $d(n) = \sigma_0(n) = \# D(n)$. Here $\sigma_0$ is the divisor function with $k = 0$, and $D(n)$ is the set of divisors of $n$.

Remarks

Other common notations are $\tau(n)$ and $\sigma_0(n)$. If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$, then $d(n) = (e_1 + 1) \cdots (e_r + 1)$. For example, $12 = 2^2 \cdot 3$, so $d(12) = 3 \cdot 2 = 6$. The number $d(n)$ is odd if and only if $n$ is a perfect square.

Explanations

L1 human

The function $d(n)$ counts the divisors of $n$. A divisor of $p^e q^f$ is $p^a q^b$ with $0 \le a \le e$ and $0 \le b \le f$. So there are $(e + 1)(f + 1)$ choices.