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- W3033707919 abstract "Abstract Let $(F_{n})_{ngeq 0}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> be the Fibonacci sequence. The order of appearance $z(n)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>z</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:math> of a positive integer n is defined as $z(n):= min {kgeq 1: nmid F_{k}}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>z</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mo>min</mml:mo><mml:mo>{</mml:mo><mml:mi>k</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mi>n</mml:mi><mml:mo>∣</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:math> . In 2013, Marques proved that $liminf_{nto infty }z(n)/n=0$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mo>lim inf</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:msub><mml:mi>z</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math> . Let ϵ be a positive real number. In this paper, in particular, we generalized this Marques’ result by proving that almost all positive integers satisfy $z(n)/n<epsilon $ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>z</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo><</mml:mo><mml:mi>ϵ</mml:mi></mml:math> ." @default.
- W3033707919 created "2020-06-12" @default.
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- W3033707919 date "2020-06-05" @default.
- W3033707919 modified "2023-09-26" @default.
- W3033707919 title "Some problems related to the growth of $z(n)$" @default.
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- W3033707919 doi "https://doi.org/10.1186/s13662-020-02735-5" @default.
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