Reference densities from accurate configuration interaction wave functions for the beryllium isoelectronic series were used to solve the Kohn-Sham equations using a constrained search that minimizes the kinetic energy. For Z<25 in the series, a single Kohn-Sham determinant is sufficient to give the minimum kinetic energy. For higher Z a single Kohn-Sham determinant produces 2p eigenvalues that are lower than the 2s eigenvalues, and a kinetic energy that is not the minimum that can be obtained from an antisymmetric wave function that produces the reference density. Fractional occupation numbers are required to obtain the minimum kinetic energy, and at the minimum kinetic energy the 2s and 2p eigenvalues become equal. Values of the optimal 2p occupation numbers approach 0.09 for high Z.