Consider \[f(n)=n+ \left[ \frac{n}{6} \right] -\left[ \frac{n}{2} \right] - \left[ \frac{2n}{3} \right]\]Then \begin{align*} f(n+6)&=n+6+ \left[ \frac{n+6}{6} \right]-\left[ \frac{n+6}{2} \right] - \left[ \frac{2n+12}{3} \right]\\ &=n+ \left[ \frac{n}{6} \right] -\left[ \frac{n}{2} \right] - \left[ \frac{2n}{3} \right]+7-7=f(n) \end{align*}so we only consider modulo $6$. We see that the only ones that work are $1$, so the answers are $6k+1$.