In the Generalized Sturmian Method, solutions to the many-particle
  Schr\"odinger equation are built up from isoenergetic sets of
  solutions to an approximate Schr\"odinger equation with a weighted
  potential $\beta_\nu \op{V}_0(\xx)$. The weighting factors
  $\beta_\nu$ are chosen in such a way as to make all of the members
  of the basis set correspond to the energy of the state being
  represented. In this paper we apply the method to core ionization in
  atoms and atomic ions, using a basis where $\op{V}_0(\xx)$ is chosen
  to be the nuclear attraction potential. We make use of a large-$Z$
  approximation, which leads to extremely simple closed-form
  expressions not only for energies, but also for values of the
  electronic potential at the nucleus. The method predicts
  approximately piecewise linear dependence of the core-ionization
  energies on the number of electrons $N$ for isonuclear series, and
  an approximately linear dependence of $\Delta E-Z^2/2$ on the
  nuclear charge $Z$ for isoelectronic series.