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Statement

For a sequence $A=\{a_1<a_2<\cdots\}$ with property P (no member divides a sum of two members larger than itself) the paper states, as beliefs rather than theorems:

1. (p. 97, display (1)) "We believe that if $A$ has property P then $\max A(x)=[\tfrac13x]+1$", the maximum over finite sets of positive integers not exceeding $x$; "To show that $\max A(x)\ge[\tfrac13x]+1$ is easy---it suffices to let $A$ be the $[\tfrac13x]+1$ greatest integers not exceeding $x$" (p. 98). The paper adds that Szemerédi proved (oral communication) that $A(x)>[\tfrac13x]+1$ forces three distinct terms $a_i,a_j,a_l$ with $a_i\mid a_j+a_l$ and $(a_j+a_l)/a_i\ne2$.
2. (p. 98) "Probably, if $A$ satisfies P then $\sum1/a_i$ is convergent and in fact $\sum1/a_i<c$ where $c$ is an absolute constant."
3. (p. 98) "Also, probably, $A(x)<x^{1-c_1}$ for infinitely many $x$."

The example on p. 98: $a_i=p_i^2$ with $p_i$ the $i$-th prime congruent to $3$ modulo $4$ has property P and $A(x)>cx^{1/2}/\log x$ for every $x$; "We have not been able to do better."

Source. P. Erdős and A. Sárközi, On the divisibility properties of sequences of integers, Proc. London Math. Soc. (3) 21 (1970), 97--101; printed pp. 97--98 (PDF pp. 1--2 of the five-page Rényi scan), read on the page images.

Read depth. Claims checked: the displayed conjecture (1), the two sentences of p. 98 and the example were read clause by clause on the page images. The example's property P (a sum of two squares of primes $\equiv3\ (\mathrm{mod}\ 4)$ is not divisible by such a prime) is stated without proof in the paper and was not checked here.

Proof pointer

None; conjectures and an example. Item 1 is answered by Bedert 2023 in the reading where the two larger terms may coincide (the maximum is then $\lceil x/3\rceil$ for large $x$, one less than $[x/3]+1$ when $3\mid x$); in the paper's own reading, in which the $[\tfrac13x]+1$ greatest integers qualify, the exact maximum is not settled by that theorem. Items 2 and 3 are the third and second questions of Problem 12 in the site's order; item 3 is refuted by the site-accepted constructions of 2026 and item 2 is open.

Dependencies

None.

Bears on

• Problem 12: the origin of all three questions and of the $p^2$ example.
• Problem 13: the origin of the finite conjecture, in the paper's reading with distinct larger terms; the site's wording follows Bedert's reading.