Triple-Free Subsets

Call $S \subseteq A = \{1, \dots, 2045\}$ triple-free if no element of $S$ is three times another. Let $n$ be the number of triple-free subsets of maximum size; write $n = p^aq^b$ with $p, q$ prime. For $N = p^2 + q^2 + a^2 + b^2$, find the last three digits of $N$.