On divisibility properties of a sequence related to the quotient of fibonacci numbers

Abstract

In this study, we define the sequence (S_n) = (F_7n/13_Fn) for each positive integer n. The first few terms of this sequence are 1,29,421,8149,141961,2576099, 59831634773/13,827294629, By the divisibility properties of Fibonacci numbers, it is clear that every seventh terms of the sequence (S_n) are not an integer. We examine the divisibility properties of the sequence (S_n). We prove that (Fn,S_n) = 1 for each non- negative integer n and gcd(Sm,S_n) \ Sgcd(m,n) when m,n ≠8k, k Z+ and gcd(Sm,Fm) =gcd(S_n, Fn) = 1. Similarly, we prove that Sgcd(m,n) I gcd(Sm, S_n) when m, n j-- 8k, k 6 Z+ and u₇(m) = v₇(n). Also, we prove that Sm \ S_n when 8k, k e Z+, m \ n and u₇(m) = u₇(n). We define a subsequence (Qk(n)) of the sequence S_n as follows: Q1(n) = S_n and Qk(n) = F₇nQk-1(n), where k and n are 11011 negative integers and k ≥ 2. We prove that Sk n || Qk(n) when k, n Z+. We derive explicit formulas of the quotients upon dividing Qk{n) by Sk n for k = 2 and k = 3. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.