For a natural number $n$, the standard Mobius function $\mu(n)$ is defined to be zero if the square of any prime divides $n$, and otherwise either $-1$ or $1$ according as n has respectively an odd or even number of prime factors.
Values for the squarefree cases can be expressed formally as a product of $-1$ over every prime $p$ dividing $n$. So there is a pretty obvious and natural generalisation to similar multiplicative function defined as follows for any given integer $k > 1$, in which $\xi$ denotes a primitive root of $x^k = 1$ and $ind_{n}(p)$ the maximum power of $p$ dividing $n$ :
$\mu_k(n) = 0$ if any $p^k | n$, or $\prod_{p|n}\xi^{ind_{n}(p)}$ otherwise
I wondered if functions like this would have any neat or useful properties. I couldn't see them mentioned in the Wikipedia article on the Mobius function, so perhaps not.