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We present a new method for proving non-holonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generalize several recent results, e.g., non-holonomicity of the logarithmic sequence is extended to rational functions involving log n. Moreover, we show that the sequence that arises from evaluating the Riemann zeta function at an increasing integer sequence with bounded gap lengths is not holonomic. [ABSTRACT FROM AUTHOR]