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Claim. Let $A$ be an infinite set of positive integers such that no $a\in A$ divides $b+c$ for distinct $b,c\in A$ with $b,c>a$. Such a set contains neither $1$ nor $2$.

1. Let $P_2$ be the set of primes dividing at least two elements of $A$. Then

$$\sum_{a\in A}\frac1a\le9.33\prod_{p\in P_2}\frac p{p-1},$$

so $\sum_{a\in A}1/a<\infty$ whenever $\sum_{p\in P_2}1/p<\infty$. In particular $\sum_{a\in A}1/a\le9.33$ when the elements of $A$ are pairwise coprime, and $\sum_{a\in A}1/a\le9.33\prod_{p\le G}p/(p-1)$ when no prime above $G$ divides two elements, for instance when every two elements have greatest common divisor at most $G$. 2. Let $c_1=\min A$, and let $c_{k+1}$ be the least element of $A$ above $c_k$ that is coprime to $c_1\cdots c_k$. If this sequence is infinite, then

$$\sum_{a\in A}\frac1a\le6+\sum_{k\ge1}\frac{\log c_{k+1}}{2^k}.$$

3. For every $N\ge3$, every subset of $\{1,\ldots,N\}$ with the same property has reciprocal sum at most $1440\log\Delta_N$, where $\Delta_N=\lceil8(\log_2N+1)\rceil$. Consequently $\sum_{a\in A,\,a\le N}1/a=O(\log\log N)$, and $\sum_{a\in A}1/(a(\log\log a)^{1+\varepsilon})<\infty$ for every $\varepsilon>0$.

Covers. The third question of Problem 12, answered yes for the sets of items 1 and 2: pairwise coprime sets, sets whose pairwise greatest common divisors are bounded, and more generally sets whose shared primes have convergent reciprocal sum, together with sets whose greedy coprime elements recur as item 2 requires. For pairwise coprime sets, Schoen and Baier answer the second question; item 1 answers the third in that class. Item 3 bounds the partial reciprocal sums of every such set by $O(\log\log N)$, against the $o(\log N)$ that density zero gives, but does not give convergence. The third question for general sets remains open, so no part is settled; the first two questions are not addressed.

Credit. The analytic large sieve is used in the form of Lemma 1 of Schoen's paper, whose source card holds the bibliography, and the Brun–Titchmarsh inequality in the form of Montgomery and Vaughan's Theorem 3.9 with the error term removed as their notes state; both are used at their published statements. The residue-occupancy bound, the propositions behind items 1 and 2, and the sieve argument behind item 3 are Plasma AI's. The pairwise coprime bound was not compared with literature beyond the sources the lead cites.

Standing. Author-recorded at tier 0, with no independent review. The lead's evidence program checks the explicit constant in the pairwise coprime bound and finite instances of the bounds of item 1; items 2 and 3 and the proofs of item 1 are written arguments. Catalog acceptance is pending.

Depends on. The composite-modulus sieve lead (tier 0).