In this talk, I will first explain the concepts of Apery table of a numerical semigroup introduced by Cortedellas and Zarzuela (Tangent cones of numerical semigroup rings. Contemp. Math. 502, 45–58 (2009)) and the microinvariants of a local ring introduced by Juan Elias (On the deep structure of the blowing-up of curve singularities. Math. Proc. Camb. Philos. Soc. 131, 227–240 (2001)). We use these concepts to give some partial results about a conjecture on the regularity index of a local ring and to give some open problems.