This study theoretically proposes a logic gate for sequenced digital data with a single nonlinear nanomechanical resonator. Nonlinear vibration with cusp-type bifurcation provides us a vibrational ``memory effect\'\' that can enable logical computations for sequenced data. Our well-designed sequential force drives the resonator and can perform various logical computation, including OR and NAND. Whereas previous studies used multiple resonators for computations, our method requires only a single resonator. Further, operating this resonator close to the critical point enables computations even with weak signals. Because the NAND operation can represent any type of logical operation, the proposed method enables any logical calculation using nanomechanical systems for the sequenced data.