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

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F21%3A50018284" target="_blank" >RIV/62690094:18470/21:50018284 - isvavai.cz</a>
Result on the web
<a href="https://www.mdpi.com/2227-7390/9/16/1931" target="_blank" >https://www.mdpi.com/2227-7390/9/16/1931</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.3390/math9161931" target="_blank" >10.3390/math9161931</a>

Result language
angličtina
Original language name
On Some Properties of the Limit Points of (z(n)/n)(n)
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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