Lifetimes of low-lying excited states in the νi13/2+ bands of the neutron-deficient osmium isotopes 169,171,173Os have been measured for the first time using the recoil-distance Doppler shift and recoil-isomer tagging techniques. An unusually low value is observed for the ratio B(E2; 21/2+ →17/2+)/B(E2; 17/2+ → 13/2+) in 169Os, similar to the “anomalously” low values of the ratio B(E2; 41+ → 21+)/B(E2; 21+ → 0+gs) previously observed in several transitional rare-earth nuclides with even numbers of neutrons and protons, including the neighbouring 168,170Os. Furthermore, the evolution of B(E2; 21/2+ → 17/2+)/B(E2; 17/2+ → 13/2+) with increasing neutron number in the odd-mass isotopic chain 169,171,173Os is observed to follow the same trend as observed previously in the even-even Os isotopes. These findings indicate that the possible quantum phase transition from a seniority conserving structure to a collective regime as a function of neutron number suggested for the even-even systems is maintained in these odd-mass osmium nuclei, with the odd valence neutron merely acting as a “spectator”. As for the even-even nuclei, the phenomenon is highly unexpected for nuclei that are not situated near closed shells.