In this paper, we derive the average pairwise error probability (PEP) of an integer carrier frequency-offset (CFO) estimator with consistent pilots in orthogonal frequency-division multiplexing (OFDM) systems and address several issues based on PEP analysis. In particular, the relationship between the PEP and consistent pilots is established in terms of a diversity gain and a shift gain. Based on the observations, we present new criteria for sequence designs. Simulation results show that the sequence developed from these criteria yields much reduced outliers compared with conventional sequences for consistent CFO estimation in frequency-selective fading channels.