  Statistical properties of photon count maps have recently been proven as a
new tool to study the composition of the gamma-ray sky with high precision. We
employ the 1-point probability distribution function of 6 years of Fermi-LAT
data to measure the source-count distribution dN/dS and the diffuse components
of the high-latitude gamma-ray sky as a function of energy. To that aim, we
analyze the gamma-ray emission in five adjacent energy bands between 1 GeV and
171 GeV. It is demonstrated that the source-count distribution as a function of
flux is compatible with a broken power law up to energies of ~50 GeV. The index
below the break is between 1.95 and 2.0. For higher energies, a simple
power-law fits the data, with an index of 2.2^{+0.7}_{-0.3} in the energy band
between 50 GeV and 171 GeV. Upper limits on further possible breaks as well as
the angular power of unresolved sources are derived. We find that point-source
populations probed by this method can explain 83^{+7}_{-13}% (81^{+52}_{-19}%)
of the extragalactic gamma-ray background between 1.04 GeV and 1.99 GeV (50 GeV
and 171 GeV). The method has excellent capabilities for constraining the
gamma-ray luminosity function and the spectra of unresolved blazars.
