Exact divisibility properties of some subsequences of fibonacci numbers

For each positive integer n, we consider the following sequence of numbers F(n), F(nF(n)), F(nF(nF(n))), ⋯, where F(n) is the nth Fibonacci number defined in the usual way. Let G_k(n) be the kth term of this sequence. We prove that F(n)^k || G_k(n) for all positive integers k and n with n > 3. For the first nontrivial case when n = 3, we prove that F(3)^2k-1 || G_k(3) for all positive integers k. We also provide an alternative proof of the divisibility of G_k(n) by F(n)^k first proved by two authors of this work. Finally, we give explicit formulas of the quotients obtained from dividing G_k(n) by F(n)^k for the cases when k = 2 and k = 3.