  We uncover a systematic structure in the single particle phase-diagram of the
quasiperiodic Aubry-Andr\'e-Harper(AAH) model with power-law hoppings ($\sim
\frac{1}{r^\sigma}$) when the quasiperiodicity parameter is chosen to be a
member of the `metallic mean family' of irrational Diophantine numbers. In
addition to the fully delocalized and localized phases we find a co-existence
of multifractal (localized) states with the delocalized states for $\sigma<1$
($\sigma>1$). The fraction of delocalized eigenstates in these phases can be
obtained from a general sequence, which is a manifestation of a mathematical
property of the `metallic mean family'. The entanglement entropy of the
noninteracting many-body ground states respects the area-law if the Fermi level
belongs in the localized regime while logarithmically violating it if the
Fermi-level belongs in the delocalized or multifractal regimes. The prefactor
of logarithmically violating term shows interesting behavior in different
phases. Entanglement entropy shows the area-law even in the delocalized regime
for special filling fractions, which are related to the metallic means.
