One of the recursive formulas relating values of mobious function is
$$\sum_{m \leq n} \left\lfloor \dfrac {n}{m} \right\rfloor  \mu (m) = 1.$$
But inorder to find the
 $\mu(n)$ we need to know the mobious values for $m < n $. Hence
$$\mu (n) =1-\sum_{m < n} \left \lfloor \dfrac {n}{m}\right \rfloor \mu (m) .$$
Here we are dividing $n$ by the smaller  positive integers $m<n$, we don't have to know if they are factors of $n$ when $m$ has a square factor! ( $\mu(m)=0$) ,But still we have to know the factors of $m$ to conclude this!!
Refer to this paper: https://projecteuclid.org/euclid.mjms/1513306829 for the proof of the formula.