In this paper, a complexity-effective integer carrier frequency offset (IFO) and primary synchronization signal (PSS) detection scheme is proposed for 5G new radio systems. To estimate the IFO without any prior information about the PSS, the proposed detector uses the fact that three PSS sequences are cyclic-shifted versions of each other, which enables sequential detection of IFO and PSS. Both theoretical and simulation results are presented to verify the effectiveness of the proposed synchronization scheme. It is shown that the proposed scheme achieves a considerable complexity reduction while suffering from only slight degradation in the detection performance, compared to the conventional scheme.