The generation of terahertz waves from air-plasma induced by femtosecond three-color harmonic pulses with a frequency ratio of 1:2:m (m is a positive integer), based on the transient photocurrent model and the sawtooth-like electric field formed via multi-color harmonic pulses superposition, has been theoretically investigated. It can be seen that when the air is saturated ionized and the electron density reaches the same maximum, for the same number of harmonic pulses, terahertz conversion efficiency is not always higher when the electric field shape in the composed pulse envelope is closer to a sawtooth waveform and more asymmetric. Besides, the specific wavelength combination schemes of femtosecond three-color harmonic pulses with the frequency ratios of 1: 2:3 and 1:2:4 have also been simulated, which can significantly enhance the generation of terahertz waves, and are realized by adding only a set of optical parametric amplifiers on the basis of the conventional two-color laser pulse case at the frequency ratio of 1:2. Our study will be helpful to obtain intense terahertz sources and provide guidance for experimental operations.