Sets of integers with nonlong arithmetic progressions generated by the greedy algorithm

Let $ {S_k}$k terms, generated by the greedy algorithm. A heuristic formula, supported by computational evidence, is derived for the asymptotic density of $ {S_k}$k is composite. This formula, with a couple of additional assumptions, is shown to imply that the greedy algorithm would not maximize $ {\Sigma _{n \in S}}1/n$S with no arithmetic progression of k terms. Finally it is proved, without relying on any conjecture, that for all $ \varepsilon > 0$ which are less than n is greater than $ (1 - \varepsilon )\sqrt {2n} $n.