On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)_<i>n</i>

Abstract

Let (Fn)n≥0 be the sequence of Fibonacci numbers. The order of appearance of an integer n≥1 is defined as z(n):=min{k≥1:n∣Fk}. Let Z′ be the set of all limit points of {z(n)/n:n≥1}. By some theoretical results on the growth of the sequence (z(n)/n)n≥1, we gain a better understanding of the topological structure of the derived set Z′. For instance, {0,1,32,2}⊆Z′⊆[0,2] and Z′ does not have any interior points. A recent result of Trojovská implies the existence of a positive real number t2 such that Z′∩(t,2) is the empty set. In this paper, we improve this result by proving that (127,2) is the largest subinterval of [0,2] which does not intersect Z′. In addition, we show a connection between the sequence (xn)n, for which z(xn)/xn tends to r>0 (as n→∞), and the number of preimages of r under the map m↦z(m)/m.

Keywords

• order of appearance
• fibonacci numbers
• derived set
• greatest prime factor
• natural density