tellefunksjonen

Tellefunksjonen, often translated as the counting function, is a fundamental concept in number theory. It is denoted by the Greek letter tau, $\tau(n)$, or sometimes as $d(n)$. The tellefunksjonen of a positive integer $n$ is defined as the number of positive divisors of $n$. For example, the divisors of 6 are 1, 2, 3, and 6, so $\tau(6) = 4$.

The tellefunksjonen is a multiplicative function. This means that if $a$ and $b$ are coprime positive integers (i.e., their greatest common divisor is 1), then $\tau(ab) = \tau(a) \tau(b)$. This property is crucial for calculating the tellefunksjonen of any integer.

To calculate $\tau(n)$ for any $n$, one can first find the prime factorization of $n$. If $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$, where $p_1, p_2, \ldots, p_k$ are distinct prime numbers and $e_1, e_2, \ldots, e_k$ are their positive integer exponents, then the number of divisors is given by the formula $\tau(n) = (e_1+1)(e_2+1)\cdots(e_k+1)$. For instance, the prime factorization of 12 is $2^2 \cdot 3^1$. Therefore, $\tau(12) = (2+1)(1+1) = 3 \cdot 2 = 6$. The divisors of 12 are 1, 2, 3, 4, 6, and 12.

The value of the tellefunksjonen is 1 for $n=1$. For any prime number $p$, $\tau(p) = 2$, as its only divisors are 1 and $p$. The tellefunksjonen is a widely studied function in number theory due to its connection to divisibility and the structure of integers.