  We have measured the superconducting critical temperature (Tc) and the
diamagnetic susceptibility of La2-xSrxCuO4 single crystals in various magnetic
fields. We observed a field-induced evolution from an apparent Tc phase to an
intrinsic Tc1 = 15 K or Tc2 = 30 K phase characterized by "magic" hole
concentration which is commensurate with that of a two dimensional electronic
lattice. The onset Tc of the intrinsic superconducting phases remains robust up
to H = 5 Tesla. We suggest that the intrinsic superconducting phases at "magic"
doping concentrations are the pristine electronic phases of high temperature
superconductivity.
