  First-principles calculations are performed to investigate the electronic
states of 1/1 and 2/1 approximants with the composition
Al$_{15}$Zn$_{40}$Mg$_{45}$ which is close to the quasicrystal (QC)
Al$_{14.9}$Zn$_{41.0}$Mg$_{44.1}$ in which the superconductivity is recently
discovered. The density of states for the 1/1 approximant shows a wide
pseudogap structure near the Fermi level as commonly observed in various
approximants of the QCs, whereas those for the 2/1 approximant do not show such
a wide pseudogap. Instead of the wide pseudogap, the 2/1 approximant shows a
remarkable narrow pseudogap at the Fermi level in contrast to the 1/1
approximant, which shows a shallow hump at the Fermi level within the wide
pseudogap. This seems to be consistent with the experimental observation of the
electrical resistivity, which increases with decreasing temperature from the
room temperature down to around the superconducting transition temperature for
the 2/1 approximant together with the QC, whereas it monotonically decreases
for the 1/1 approximant.
