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Statement

Setting (p. 431). $A$ is an infinite sequence of integers $a_1<a_2<\cdots$, and a chain is an infinite subsequence $a_{n_1}<a_{n_2}<\cdots$ with $a_{n_i}\mid a_{n_{i+1}}$ for every $i$.

Question (5) (p. 432). The paper notes that the constant $c_3$ of Theorem 2 cannot be greater than $c_2$, and suggests that perhaps for every sequence $A$ there is a chain satisfying

$$\limsup_{y\to+\infty}\frac{1}{\log\log y}\sum_{a_{n_i}<y}1 \ge\limsup_{x\to+\infty}\frac{1}{\log\log x}\sum_{a_n<x}\frac{1}{a_n\log a_n} . \tag{5}$$

The authors write that they could neither prove nor disprove (5). As printed, (5) is stated for every sequence $A$, with no density hypothesis.

A second question (p. 435). After the proof of Theorem 2 the paper recalls a further theorem of Davenport and Erdős: under (1) there is a $k$ with $\limsup_{x\to\infty}(\log x)^{-1}\sum_{a_k\mid a_i}1/a_i>0$. It asks whether the stronger inequality

$$\limsup_{x\to+\infty}\frac{1}{\log x}\sum_{\substack{a_i<x\\ a_k\mid a_i}}\frac{a_k}{a_i} \ge\limsup_{x\to+\infty}\frac{1}{\log x}\sum_{a_k\le x}\frac{1}{a_k} \tag{19}$$

holds, and says that if (19) is true it is best possible. The print does not say how $k$ is quantified in (19).

Proof pointer

The paper proves neither (5) nor (19). For (5), Theorem 2 gives a chain whose count below $x$ exceeds $c_3\log\log x$ infinitely often with $c_3>c_2/(10c_4)$, a positive fraction of the right side of (5) when it is positive.

Read depth

Claims checked: (5), the sentence after it, and (19) with its surrounding sentences were read clause by clause on the page images of pp. 432 and 435 of the print. Nothing here is independently reviewed.

Dependencies

Theorem 2 of the same paper, for the context of (5).

Source. P. Erdős, A. Sárközi and E. Szemerédi, On divisibility properties of sequences of integers, Studia Sci. Math. Hungar. 1 (1966), 431--435; the edition read is named on the source card.

Bears on

• Problem 1217: the problem's inequality is (5). The paper states (5) for every sequence $A$; the problem asks it for sequences of positive lower logarithmic density. The paper records (5) as a question it could neither prove nor disprove.