We will investigate numerically a seesaw model with $A_4$ flavor symmetry to find allowed regions satisfying the current experimental neutrino oscillation data, then use them to predict physical consequences. Namely, the lightest active neutrino mass has order of $mathcal{O}(10^{-2})$ eV. The effective neutrino mass $|langle mrangle|$ associated with neutrinoless double beta decay is in the range of $[0.002 ;mathrm{eV},0.038;mathrm{eV}]$ and $[0.048;mathrm{eV},0.058;mathrm{eV}]$ corresponding to the normal and the inverted hierarchy schemes. Other relations among relevant physical quantities are shown, so that they can be determined if some of them are confirmed experimentally. The recent data of the baryon asymmetry of the Universe ($eta_B$) can be explained via leptogenesis caused by the effect of the renormalization group evolution on the Dirac Yukawa couplings, provided the right handed neutrino mass scale $M_0$ is ranging from $mathcal{O}(10^8)$ GeV to $mathcal{O}(10^{12})$ GeV for $tanbeta =3$. This allowed $M_0$ range distinguishes with the scale of $mathcal{O}(10^{13})$ GeV concerned by other effects that also generate the consistent $eta_B$ from leptogenesis. The branching ratio of the decay $ mu rightarrow,egamma$ may reach the future experimental sensitivity in the very light values of $M_0$. Hence, it will be inconsistent with the $M_0$ range predicted from the $eta_B$ data whenever this decay is detected experimentally.
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