Divisoranzahl

Divisoranzahl, often denoted by the function $\tau(n)$ or $d(n)$, represents the number of positive divisors of a positive integer $n$. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12, so the divisoranzahl of 12 is 6.

The divisoranzahl can be calculated using the prime factorization of $n$. If the prime factorization of $n$ is given by $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, where $p_1, p_2, \ldots, p_k$ are distinct prime numbers and $a_1, a_2, \ldots, a_k$ are their positive integer exponents, then the divisoranzahl of $n$ is given by the formula $\tau(n) = (a_1 + 1)(a_2 + 1) \cdots (a_k + 1)$.

This formula arises because any divisor of $n$ must be of the form $p_1^{b_1} p_2^{b_2} \cdots p_k^{b_k}$, where $0 \le b_i \le a_i$ for each $i$ from 1 to $k$. For each prime factor $p_i$, there are $a_i + 1$ possible choices for its exponent $b_i$ (from 0 to $a_i$). Since the choices for each prime factor are independent, the total number of divisors is the product of the number of choices for each exponent.

The divisoranzahl function is a multiplicative function, meaning that if $m$ and $n$ are coprime positive integers, then $\tau(mn) = \tau(m)\tau(n)$. The value $\tau(1)$ is 1, as 1 is the only divisor of 1. For any prime number $p$, $\tau(p) = 2$ because its only divisors are 1 and $p$.