In flexible bandwidth optical networks, the dynamic lightpath setup and teardown inevitably lead to spectrum fragmentation which blocks the new connections. In order to effectively allocate the spectrum along a better route, a notion of available spectrum consecutiveness is introduced and three dynamic routing and spectrum assignment (RSA) algorithms are proposed in this letter accordingly. The novel algorithms retain the spectrum consecutiveness as much as possible when establishing a lightpath and reduce the spectrum fragmentation. Simulation results indicate that our proposed spectrum consecutiveness based RSA algorithms achieve lower blocking probability and higher adaptability to more line rates mixture.