Distinguishing sequential pattern can be used to present the difference between data sets, and thus has wide applications, such as commodity recommendation, user behavior analysis and power supply predication. Previous algorithms on mining distinguishing sequential patterns ask users to set both positive and negative support thresholds. Without sufficient prior knowledge of data sets, it is difficult for users to set the appropriate support thresholds, resulting in missing some significant contrast patterns. To deal with this problem, an algorithm, called kDSP-miner (top-k distinguishing sequential patterns with gap constraint miner), for mining top-k distinguishing sequential patterns satisfying the gap constraint is proposed. Instead of setting the contrast thresholds directly, a user-friendly parameter, which indicates the expected number of top distinguishing patterns to be discovered, is introduced in kDSP-miner. It makes kDSP-miner easy to use, and its mining result more comprehensible. In order to improve the efficiency of kDSP-miner, several pruning strategies and a heuristic strategy are designed. Furthermore, a multi-thread version of kDSP-miner is designed to enhance its applicability in dealing with the sequences with high dimensional set of elements. Experiments on real world data sets demonstrate that the proposed algorithm is effective and efficient.