We have investigated the growth of self-catalyzed GaAs/GaAs(x)Sb(1-x) core-shell nanowires directly on Si(111) substrates by molecular beam epitaxy. The compositions of the GaAs(x)Sb(1-x) shells are tuned in a wide range where the Sb-content is varied from 10 to ~70%, covering the miscibility gap. In addition, the GaAs(x)Sb(1-x) shells are grown on both zinc blende (ZB) and wurtzite (WZ) crystal structures. Morphological and structural characterizations of the grown nanowires indicate successful transfer of the GaAs core crystal structure to the GaAs(x)Sb(1-x) shells for both ZB and WZ nanowires, with slower shell growth rate on the WZ segments.