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

Abstract

Let (F-n)(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 vertical bar 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}subset of Z' subset of [0,2] and Z' does not have any interior points. A recent result of Trojovska implies the existence of a positive real number t < 2 such that Z' boolean AND (t,2) is the empty set. In this paper, we improve this result by proving that (12/7,2) is the largest subinterval of [0,2] which does not intersect Z'. In addition, we show a connection between the sequence (x(n))(n), for which z(x(n))/x(n) tends to r > 0 (as n -> infinity), and the number of preimages of r under the map m -> z(m)/m.