piecewise defining confusion

Question

I am trying to define a recurrence relation with a piecewise and I was wondering whether I could set my one of my if conditions to be $\mathcal{S}(n) = \mathcal{S}(n-4) + 1 \hspace{.3cm}$ if $\hspace{.3cm} \frac{n}{6} = m \hspace{0.3cm}(where \hspace{0.3cm} m \in \mathbb{N})$. Is this a valid if condition? Further is there a way so that I can have $\frac{n}{6} = m_{1} \hspace{0.3cm} (m_{1} \in \mathbb{N})$ but not have $\frac{n}{10} = m_{2} $ where $m_{2} \notin \mathbb{N}$?