Erdős Problem #858 — Let $A\subseteq \{1,\ldots,N\}$ be such that there is no solution to $at=b$ with $a,b\in A$ and the smallest prime factor of $t$ is $>a$.

Erdős posed this weighted finite extremal problem in 1970: estimate the largest value of the normalized reciprocal sum over admissible subsets of $\{1,\ldots,N\}$.

Known results

• Alexander (1966) proved that admissible infinite sets have reciprocal sums $o(\log N)$.
• Erdős, Sárközi, and Szemerédi (1968) proved the same normalized upper bound for the corresponding infinite-set formulation.
• A 2022 study of $L$-primitive sets gives an exact supremum for $\sum_{a\in A}1/(a\log a)$ and proves a tail bound with limit $e^\gamma$, but does not state the finite-$N$ maximum in precisely the problem’s normalization.

Undated claimed resolution; related 2022 paper

An online paper claims a sharp finite frontier theorem and $M(N)=(c_2+o(1))\log N$, with $c_2=0.6187712111099834\ldots$ and cutoff exponent $\alpha_2=0.2804383098923534\ldots$. The problem entry attributes this claim to Chojecki and GPT-5.4 Pro, but the result remains unverified; no retrieved source reports a subsequent correction or independent confirmation.

Current status (as of September 2026): The classical $o(\log N)$ bound is known, while the claimed sharp finite-$N$ asymptotic remains unverified.