If $ a < n-3$ is a composite odd integer that is congruent to $ n$ modulo $ 3$, then $ n-a = 3k$ is also a composite integer. Now let's have a look here: http://www.research.att.com/~njas/sequences/A000040. Case 1. $ n\equiv 0\left(\bmod\, 3\right)$ $ a = 9$ works, so all $ n > 12$ are good. Case 2. $ n\equiv 1\left(\bmod\, 3\right)$ $ a = 25$ works, so all $ n > 28$ are good. Case 3. $ n\equiv 2\left(\bmod\, 3\right)$ $ a = 35$ works, so all $ n > 38$ are good. The answer is $ \boxed{38}$ (if my computations are correct).