## Abstract

In this note, we prove that there is no transcendental entire function f(z) ∈ ℚ[[z]] such that f(ℚ) ⊆ ℚ and den f(p/q) = F(q), for all sufficiently large q, where F(z) ∈ ℤ[z].

In this note, we prove that there is no transcendental entire function f(z) ∈ ℚ[[z]] such that f(ℚ) ⊆ ℚ and den f(p/q) = F(q), for all sufficiently large q, where F(z) ∈ ℤ[z].

Let F_{n} be the nth Fibonacci number and let L_{n} be the nth Lucas number. The order of appearance z(n) of a natural number n is defined as the smallest natural number k such that n divides Fk. For instance, z(F_{n}) = n = z(L_{n})/2 for all n > 2. In this paper, among other things, we prove that

for all positive integers n ≡ 0,8 (mod 12).