In this work we use DEM simulations to fnd the densest packing of frictionless spheres subject to mechanical compression by changing systematically the grain size distribution (GSD). This is done by modeling the GSD as a power law and varying the GSD describing parameters: the size span and the distribution shape, while relating these parameters to the density and other micro-structural parameters. For large size spans, the optimal GSD resembles the century old Fuller-Thomson distribution that is commonly used in the feld of cements and pavements. This optimal power-law GSD produces not only the densest packing but also presents good connectivity, measured from a small proportion of foating particles, and reduces the packing internal spatial correlations, measured by the pair correlation function. Thus, tuning the GSD appropriately allows for designing mechanically stable packings that have the highest density, a small number of foating particles, and less local correlations.