On Some Properties of the Limit Points of (z(n)/n)n

Mathematics

Kandasamy Venkatachalam¹  Eva Trojovská¹ 

[1] Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic;

关键词: order of appearance;    fibonacci numbers;    derived set;    greatest prime factor;    natural density;   

DOI  :  10.3390/math9161931

来源: DOAJ

【 摘 要 】

Let ${\left(F_n\right)}_{n\ge 0}$ be the sequence of Fibonacci numbers. The order of appearance of an integer $n\ge 1$ is defined as $z\left(n\right):=min\{k\ge 1:n\mid F_k\}$. Let ${\mathcal{Z}}^{\prime }$ be the set of all limit points of $\left\{z\right(n)/n:n\ge 1\}$. By some theoretical results on the growth of the sequence ${(z\left(n\right)/n)}_{n\ge 1}$, we gain a better understanding of the topological structure of the derived set ${\mathcal{Z}}^{\prime }$. For instance, $\{0,1,\frac{3}{2},2\}\subseteq {\mathcal{Z}}^{\prime }\subseteq [0,2]$ and ${\mathcal{Z}}^{\prime }$ does not have any interior points. A recent result of Trojovská implies the existence of a positive real number $t<2$ such that ${\mathcal{Z}}^{\prime }\cap (t,2)$ is the empty set. In this paper, we improve this result by proving that $(\frac{12}{7},2)$ is the largest subinterval of $[0,2]$ which does not intersect ${\mathcal{Z}}^{\prime }$. In addition, we show a connection between the sequence ${\left(x_n\right)}_n$, for which $z\left(x_n\right)/x_n$ tends to $r>0$ (as $n\to \infty$), and the number of preimages of r under the map $m\mapsto z\left(m\right)/m$.

【 授权许可】

Unknown