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 $>a$ . Estimate the maximum of

\frac{1}{\log N}\sum_{n\in A}\frac{1}{n}.

Source: erdosproblems.com/858

Number theory
Primitive sets

Wiki page

Status

SolvedAnswered

An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved. The site credits the solution to Chojecki and GPT-5.4 Pro; see Chojecki's asymptotic constant.

Al66: Alexander, Ralph, Density and multiplicative structure of sets of integers. Acta Arith. 12 (1967), 321-332.
Be35: Behrend, F., On sequences of numbers not divisible by another. London Math. Soc. Journal (1935), 42-45.
ESS68: Erdős, P. and Sárközi, A. and Szemerédi, E., On the solvability of certain equations in sequences of positive upper logarithmic density. J. London Math. Soc. (1968), 71-78.
Er70: Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133; the problem is on p. 128. Library home: Erdos 1970.

Problem 858

Status

Proof claims

Discussion