In the present paper, we analyze the problem of an arbitrary number of particlesAin a harmonic-oscillator common potential interacting through a harmonic-oscillator force. The particles have spin12and obey Fermi statistics. The exact lowest eigenstate and energy are derived for values ofAcorresponding to closed subshells. An approximate analysis is also made using determinantal wave functions formed from single-particle harmonic-oscillator states, giving the eigenstates and energies of a simple form of the Hartree-Fock approximation. The exact and Hartree-Fock energies and eigenstates are compared both when potential and force have the same sign, and when they have opposite sign. We discuss the results for models which we call pseudonuclei and pseudoatoms, and we indicate some of the inferences for the real thing.