We present a systematic study of filamentary ultrashort laser pulses in air, through numerical solutions of the nonlinear Schrödinger equation for various contributions of the delayed Kerr nonlinearity. The results show that a relatively larger contribution of the delayed Kerr nonlinearity will lead to a longer stable filament. This is explained from the transfer of the nonlinear contributions from the frontier part to the back of the pulse and the counterbalanced action of the negative plasma induced nonlinearity by the delayed Kerr nonlinearity in the trailing part of the pulses. Furthermore, effect of ionization on the stability of the filament is investigated. Two formulas are used to generate the data of the ionization, i.e., the Perelemov, Popov, and Terent'ev (PPT) and the Ammosov, Delone, and Krainov formula. It is found that simulation with higher ionization rate (PPT) could generate a more stable and longer filament.