[show abstract][hide abstract] ABSTRACT: We present the first study on the mass functions of Jenkins et al (J01) and an estimate of their corresponding largest virialized dark halos in the Universe for a variety of dark-energy cosmological models with a running spectral index. Compared with the PL-CDM model, the RSI-CDM model can raise the mass abundance of dark halos for small mass halos at lower redshifts, but it is not apparent on scales of massive mass halos. Particularly, this discrepancy increases largely with the decrease of redshift, and the RSI-CDM model can suppress the mass abundance on any scale of halo masses at higher redshift. As for the largest mass of virialized halos, the spatially flat $\Lambda$CDM models give more massive mass of virialized objects than other models for both of PL-CDM and RSI-CDM power spectral indexs, and the RSI-CDM model can enhance the mass of largest virialized halos for all of models considered in this paper. So we probably distinguish the PL-CDM and RSI-CDM models by the largest virialized halos in the future survey of cluster of galaxies.