We demonstrate pure wurtzite InAs nanowires over a wide diameter range. The nanowires are grown on an InP (111)B substrate by Au-catalyzed metal organic chemical vapor deposition. The crystal structure of nanowires strongly depends on the growth temperature and V/III ratio. Under optimized growth condition, pure wurtzite phase is obtained over the full length of the tens-of-micrometer-long nanowire, covering a broad diameter range of 70-420 nm. The broad diameter range of wurtzite phase is attributed to a high actual V/III ratio due to a reduction of In amount because of the InP substrate. The pure wurtzite nanowires exhibit much higher electron mobility than those with dense stacking faults, showing great potential in high-speed electronic and optoelectronic devices.