For main-sequence G and K stars we study again the empirical relations between the periods of the activity cycles, Pcyc, and the rotational periods, prot. We use the high-quality data selected by Brandenburg, Saar, and Turpin. As found by those authors ``the Pcyc increase proportional to the prot, along two distinctly different sequences,'' the active ``A'' sequence, and the inactive ``'I'' sequence with cooler and more slowly rotating stars. It is found here that along each sequence the number of rotation periods per activity cycle is nearly the same, but the numbers are different for the different sequences, indicating that probably different kinds of dynamos are working for the stars on the different sequences. The transition from one sequence to the other occurs at a rotation period of 21 days. The rotation periods then increase abruptly by about a factor of 2 for the cooler stars. We suggest that this indicates abruptly increased deep mixing. Along the I sequence the overall dependence of the Ca II emission line fluxes, F(Ca II), on rotation and Teff is consistent with F(Ca II)~T4effp-4/3rot. For the A-sequence stars the dependence of F(Ca II) on rotation seems to be stronger than for the I-sequence stars.