On the increasing sequences of positive integers avoiding arithmetic progressions of specified lengths, what is the intuitive reason for jumps in the scatterplots for odd $n$, but no jumps (i.e., smooth graphs) for even $n$?
Edit: the graph for $n=9$ is smooth, so this pattern doesn't hold, and it makes me think that, in fact, the pattern is for prime numbers. Is this the case then? Whatever the case, I'm interested to know about the jumps and an explanation of when the jumps occur and why only for particular values of $n$...
The sequences and their graphs I'm referring to can be found in the links at the bottom of this OEIS page (If you go into each link, you can click the link labelled "graph" near the top of the linked page)