Evidence demonstrates that the behavior of climate, the time between R peaks of an electrocardiogram signal, or the time between firings of neurons, exhibit long-range dependence and monofractal/multifractal characteristics. The synthesis of discrete sequences with this type of dependency then requires generators of random numbers different from those used in cryptography.In the quest for a solution embodying such dependence, the authors of related researches detected, by means of simulation tools, the existence of monofractal/multifractal behavior of a memristor-based random number generator. Notwithstanding the novelty of the finding, the logical step to follow requires the implementation in hardware, to investigate the details, limitations, and difficulties, which are not contemplated in the advanced simulations. With this purpose in mind, the aforementioned research is continued, describing the components and selection criteria considered for the hardware implementation of a reference memristor and one proposed by the authors, the chaotic circuit leveraging them, and the processing that is performed on the chaotic signals to achieve the random discrete sequences. After applying the estimation tools, findings indicate that more than 60% of the proposed combinations allow generating random discrete sequences, with long-range dependence, and that both monofractal and multifractal behaviors can also be obtained. Consequently, a hardware system was achieved that can be used as a source of entropy in future synthetic biological signal generators.