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

Fraction of delocalized eigenstates in the long-range Aubry-Andr\'e-Harper model