Saturated nanotubes consisting of 2-10 and 20 layers of cyclic units of six-membered rings, each one having a pyrimidine-like framework (i.e., -C-C-C-N-C-N-), were studied by quantum chemistry methods using Density Functional Theory (DFT) at the B3LYP/6-31G* level of theory. Four different nanotube (NT) configurations were theoretically studied in this work. They were formed by covalently arranging each layer over the other, with uniform relative rotations of 0A degrees, 60A degrees, 120A degrees, and 180A degrees with respect to each of the layers. Different structures can be created by modulating the relative rotation as layers are added to the main nanostructure. NTs with a relative rotation of 60A degrees showed both greater stabilities and highest potential for catalytic activity. All of them showed band gaps of around 0.2 eV. Charges and other properties can be controlled by appropriate layer arrangement. The studied families of NTs have a very small diameter and could find potential applications in chemistry, physics, and medicine.