A number of conformers of aqua-K+ complexes, K+(H2O)(n) (n=1-10) have been investigated using high level ab initio calculations, to elucidate the structures and thermodynamic energies of the hydrated potassium ions. Since the coordination number of K+ is around six in the bulk water, the focus of the present study has been the n=5 and 6 clusters. In contrast to previous studies which have used only the enthalpies to compare against the experimental numbers, the present study also employs free energies. As a result, the predictions of a number of hitherto unknown conformers are in excellent agreement with the experimental results. The maximum coordination number for K+ in ligands containing O atoms is evaluated to be around eight from the energetics of structures possessing only the first hydration shell of water molecules around the K+ ion. It is of interest to note that the hydration of the K+ ion is less structured than that of the Na+ ion, since the water-water interaction becomes more important in the aqua-K+ clusters. The predicted vibrational frequencies of the aqua-K+ clusters reflect the H-bonding signature, and hence, could be utilized in the identification of the hydration structures of K+ in experiments.