  We report characterization results by energy dispersive x-ray analysis and
AC-susceptibility for a statistically relevant number of single layer
Bi-cuprate single crystals. We show that the two structurally quite different
modifications of the single-layered Bi-cuprate, namely (La,Pb=0.4)-Bi2201 and
La-Bi2201, exhibit anomalies in the superconducting transition temperature at
certain hole doping, e.g. at 1/8 holes per Cu. These doping values agree well
with the 'magic doping fractions' found in the temperature dependent resistance
of LSCO by Komiya et al. This new set of findings suggests that all these
anomalies are generic for the hole-doped high-temperature superconductors.
