Floor Function Returns

Let $\frac{n}{3}-\frac{1}{2}=A$

$\left\lfloor A\right\rfloor \left\lfloor A+1\right\rfloor\\=\left\lfloor A\right\rfloor \left( \left\lfloor A\right\rfloor+1\right )$

Let $\left\lfloor A\right\rfloor=k$

So, $k(k+1)$ is a prime number where k is integer.

Possible values are $1×2$ and $-2×-1$

$k=1,-2\Rightarrow A\in [-2,-1) \cup [1,2)\\\Rightarrow \frac{n}{3}\in [-1.5,-0.5) \cup [1.5,2.5)\\\Rightarrow n\in [-4.5,-1.5)\cup [4.5,7.5)$

So values of $n$ are $\{-4,-3,-2,5,6,7\}$