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Claim. Let $\rho(n)=\max\sum1/a_i$, the maximum taken over collections of integers $1<a_1<\dots<a_r$ with $[a_i,a_j]>n$ for all $1\le i<j\le r$. Then $\limsup\rho(n)\le1.0170166$, improving the constant $c=1.017262\ldots$ of Schinzel and Szekeres's Theorem 2. The paper also observes that Erdős's conjecture $\rho(n)\to1$ would follow from a real function $h$ with $h(x)=0$ for $x<1$, $h(x)\ge0$ and $S(x)=\sum_{k\ge1}h(x/k)\ge x$ for $x>1$ and $S(x)/x\to1$, and constructs a function with weaker properties, from which the constant comes. Since $1.0170166<31/30$ and the element $1$ adds nothing to the question (a set containing $1$ is $\{1\}$, with sum $1$), every admissible set has reciprocal sum below $31/30$ once $n$ is large, which answers the first question of Problem 542 yes for all sufficiently large $n$. The theorem is stated as the zbMATH review of the paper by I. Z. Ruzsa (Zbl 0870.11013) gives it. The site's commentary states the result in the explicit form $\sum1/a<1/3+1/4+1/5+1/7+1/11=1.017099\ldots$ for $n>172509$; that form is the site's and was not read in the paper. The thread's $\rho_n<1.0170166$ for large $n$ agrees with the review. The source is Y.-G. Chen, On a problem of P. Erdős, Acta Sci. Math. (Szeged) 62 (1996), no. 1--2, 101--114.

Covers. The first question for all sufficiently large $n$.

Acceptance. Refereed: Acta Scientiarum Mathematicarum (Szeged) is a refereed journal. The site's curator, Thomas Bloom, cites the result in the problem's commentary, which credits the solution of both questions to Schinzel and Szekeres. The record gives the year 1996 and no month or day, so the month and day in the page name are placeholders.

Depends on. No page of this wiki.