## Longeur maximale d’un chemin élémentaire du graphe divisoriel. (Maximal length of an elementary path of the divisor graph).(French)Zbl 0766.11039

Let $$f=f(x)$$ denote the maximum number of distinct positive integers $$n_ i$$ less than $$x$$ with the property that $$n_ i\mid n_{i+1}$$ or $$n_{i+1} \mid n_ i$$ for $$i=1,2,\dots,f-1$$. The author announces the result $f(x)\geq {x \over \log x} \exp(-(2e^{1/2}+o(1))(\log \log x)^{1/2}) \quad \text{as} \quad x\to\infty,$ and outlines its proof. This is a significant improvement on previous results, but he remarks that G. Tenenbaum has since obtained a further improvement.

### MSC:

 11N25 Distribution of integers with specified multiplicative constraints 11N45 Asymptotic results on counting functions for algebraic and topological structures

### Keywords:

divisor graph; asymptotic estimate; divisibility properties