A series of high-quality under-doped La2-xSrxCuO4 superconductor crystals with x = 0.063 - 0.125 were prepared by traveling-solvent floating-zone (TSFZ) technique. We found by dc magnetic measurements that, in this series of crystals, the superconducting transition was quite sharp in the vicinity of the hole densities of x = 1/16 and x = 1/9 while it was much broader away from these two magic numbers, and the Meissner fraction showed a remarkable minimum near x = 1/9. We concluded that these phenomena are reflections of intrinsic properties of this cuprate system. Our observations are discussed in light of recently proposed composite charge model together with charge inhomogeneity and electronic phase separations.
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