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

Eva Trojovská and Kandasamy Venkatachalam
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Eva Trojovská: Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
Kandasamy Venkatachalam: Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic

Mathematics, 2021, vol. 9, issue 16, 1-8

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 ? 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 .

Keywords: order of appearance; fibonacci numbers; derived set; greatest prime factor; natural density (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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