The antibunching effects for the eigenstates of the Nth powers (N > 3) of the annihilation operator of the non-harmonic oscillator are studied. Using the numerical method, we have investigated the influences of the non-harmonic term on the effects under the circumstance of N=3. The results show that the antibunching effects for the eigenstates of the higher powers of annihilation operator of the non-harmonic oscillator are evidently different from those of the eigenstates of the harmonic oscillator. The effects can be shown in a number of intervals alternately when x = |β|2, which reflects the intensity of non-harmonic oscillator light field, is changed, and the intervals that the antibunching effects are exhibited are changed larger when the non-harmonic term is greater.