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Inverse design of short single-stranded RNA and DNA sequences (aptamers) is the task of finding sequences that satisfy a set of desired criteria. Relevant criteria may be, for example, the presence of specific folding motifs, binding to molecular ligands, sensing properties, etc. Most practical approaches to aptamer design identify a small set of promising candidate sequences using high-throughput experiments (e.g. SELEX), and then optimize performance by introducing only minor modifications to the empirically found candidates. Sequences that possess the desired properties but differ drastically in chemical composition will add diversity to the search space and facilitate the discovery of useful nucleic acid aptamers. Systematic diversification protocols are needed. Here we propose to use an unsupervised machine learning model known as the Potts model to discover new, useful sequences with controllable sequence diversity. We start by training a Potts model using the maximum entropy principle on a small set of empirically identified sequences unified by a common feature. To generate new candidate sequences with a controllable degree of diversity, we take advantage of the model's spectral feature: an energy bandgap separating sequences that are similar to the training set from those that are distinct. By controlling the Potts energy range that is sampled, we generate sequences that are distinct from the training set yet still likely to have the encoded features. To demonstrate performance, we apply our approach to design diverse pools of sequences with specified secondary structure motifs in 30-mer RNA and DNA aptamers.