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

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 ∣ F k } . 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 , 3 2 , 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 t < 2 such that Z ′ ∩ ( 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 → ∞ ), and the number of preimages of r under the map m ↦ z ( m ) / m .