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On some sequences of integers. (English) Zbl 0015.15203
Let $$a_1 < a_2 < \cdots < a_r \leq n$$ be a set of positive integers such that $$a_i-a_j\neq a_j-a_k$$ for $$1\leq k < j < i\leq r$$. For given $$n$$ let $$r(n)$$ be the maximum value of $$r$$ for which such a set exists. The authors prove that (1) $$r(2n)\leq n$$ for $$n\geq 8$$, (2) $$\limsup \frac{r(n)}{n} \leq \frac 49$$. They conjecture that $$r(n)=o(n)$$, and G. Szekeres conjectured that $$r(\tfrac12 (3^k+1)) =2^k$$.

##### MSC:
 11B75 Other combinatorial number theory 11B05 Density, gaps, topology
##### Keywords:
sequences of integers; density; arithmetic progression
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