||This article concerns with the weak 16-th Hilbert problem. More precisely, we consider the uniform isochronous centers x'=-y x^(n-1) y, y'= x x^(n-2) y^2 , for n = 2, 3, 4, and we perturb them by all homogeneous polynomial of degree 2, 3, 4, respectively. Using averaging theory of first order we prove that the maximum number N (n) of limit cycles that can bifurcate from the periodic orbits of the centers for n = 2, 3, under the mentioned perturbations, is 2. We prove that N (4) 2, but there is numerical evidence that N (4) = 2. Finally we conjecture that using averaging theory of first order N (n) = 2 for all n > 1. Some computations have been made with the help of an algebraic manipulator as mathematica.