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Stochastic computing (SC) division circuits have gained importance in recent years compared to other arithmetic circuits due to their low complexity as a result of an accuracy tradeoff. Designing a division circuit is already complex in conventional binary-based hardware systems. Developing an accurate and efficient SC division circuit is an open research problem. Prior work proposed different SC division circuits by using multiplexers and JK-flip-flop units, which may require correlated or uncorrelated input bit-streams. This study is primarily centered on exploring a cost-effective and highly efficient bit-stream generator specifically designed for SC division circuits. In conjunction with this objective, we assess the performance of multiple bit-stream generators and analyze the impact of correlation on SC division. We compare different designs in terms of accuracy and hardware cost. Moreover, we discuss a low-cost and energy-efficient bit-stream generator via powers-of-2 Van der Corput (VDC) sequences. Among the tested sequence generators, our best results were achieved with VDC sequences. Our evaluation results demonstrate that the novel VDC-based design yields promising outputs, resulting in a 15.5% reduction in the area-delay product and an 18.05% saving in energy consumption for the same accuracy level compared to conventional bit-stream generators. Significantly, our investigation reveals that employing the proposed generator improves the precision compared to the state-of-the-art. We validate the proposed architecture with an image processing case study, achieving high PSNR and structural similarity values.