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Statement

Item 6 of the new-problems half (printed p. 150, in Hungarian): let $a_1<\dots<a_k\le n$ be an arbitrary sequence and $b_1<\dots$ the sequence of those numbers that are multiples of at least one $a$ ("azon számok sorozata, melyek legalább egy $a$-nak többszörösei"). Is it true that for every $m>n$

$$\frac{B(m)}{m}<\frac{2B(n)}{n},\qquad B(x)=\sum_{b_i\le x}1\,? \tag{1}$$

In translation: "(1), if true, clearly cannot be improved: let the sequence $a_1<\dots$ consist only of $a_1$, $n=2a_1-1$, $m=2a_1$. We may also remark that there is no $\varepsilon>0$ for which $B(m)/m>\varepsilon B(n)/n$ holds for every sequence $a_1<\dots<a_k\le n$ and $m>n$": take the $a$'s to be the numbers between $n/2$ and $n$ and $m=m(n)$ large; if $n>n_0(\varepsilon)$ then $B(m)/m<\varepsilon/2$. The item cites Erdős, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), 126--128.

So the 1966 paper defines $B$ as the count of multiples ($a\mid b$), the site's reading of Problem 488; the 1961 problem paper prints the count of non-multiples at its item (I.27.1) (printed p. 236; filed as erdos_1961_unsolved_problems).

Source. P. Erdős, Számelméleti megjegyzések, V. Extremális problémák a számelméletben, II, Mat. Lapok 17 (1966), 135--155; item 6 on printed p. 150 (PDF p. 16 of the Rényi archive's 21-page scan), read on the page image.

Read depth. Claims checked: the definition, display (1), the single-element example and the reverse-inequality remark were read clause by clause on the page image. A question with two elementary remarks; no proof.

Proof pointer

None; a question. The two remarks are elementary: for the single set $\{a_1\}$ with $n=2a_1-1$ and $m=2a_1$, $B(n)/n=1/(2a_1-1)$ and $B(m)/m=2/(2a_1)=1/a_1$, so $(B(m)/m)/(B(n)/n)=(2a_1-1)/a_1\to2$, and no constant below $2$ works in (1); the reverse example follows from the multiples of the numbers in $(n/2,n]$ thinning out.

Dependencies

None (a question).

Bears on

• Problem 488: the problem's statement in the 1966 wording with $B$ the count of multiples, and the sharpness example the site's page records; the 1961 paper's item prints the opposite divisibility condition.