The $\tau$ function, also called the divisor function, takes a positive integer as its input and gives the number of positive divisors of its input as its output. For example, since $1$ $2$ and $4$ are all of the positive divisors of $4$ we have $\tau (4)=3$ As another example, since $1$ $2$ $5$ and $10$ are all of the positive divisors of $10$ we have $\tau (10)=4$

The $\tau$ function behaves according to the following two rules:

1. If $p$ is a prime and $k$ is a nonnegative integer, then $\tau(p^k)=k+1$

2. If $\gcd(a,b)=1$ then $\tau(ab)=\tau(a)\tau(b)$

Because these two rules hold for the $\tau$ function, it is a multiplicative function.

Note that these rules work for the previous two examples. Since $2$ is prime, we have $\tau(4)=\tau(2^2)=2+1=3$ Since $2$ and $5$ are distinct primes, we have $\tau(10)=\tau(2\cdot 5)=\tau(2)\tau(5)=(1+1)(1+1)=4$

If $n$ is a positive integer, the number of prime factors of $x^n-1$ over $\mathbb{Q}[x]$ is $\tau(n)$ For example, $x^9-1=(x^3-1)(x^6+x^3+1)=(x-1)(x^2+x+1)(x^6+x^3+1)$ and $\tau(9)=3$

The $\tau$ function is extremely useful for studying cyclic rings.

The sequence $\{\tau(n)\}$ appears in the OEIS as sequence A000005.