ARITHMETIC FUNCTIONS OF FIBONACCI AND LUCAS NUMBERS

Let F-n and L-n be the nth Fibonacci and Lucas numbers, respectively. Let phi(n) be the Euler totient function of n and sigma(k)(n) the sum of kth powers of the positive divisors of n. Luca obtained the inequalities phi(F-n) >= F-phi(n), sigma(0)(F-n) >= F-sigma 0(n), and sigma(k)(F-n) <= F-sigma k(n) for all n, k >= 1. In this article, we extend Luca's result by replacing the function phi by phi(k) and J(k), which are generalizations of phi. We also consider the corresponding results for phi(k)(L-n), L-phi k(n), J(k)(L-n), L-Jk(n), sigma(k)(L-n), and L-sigma k(n).