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The core of the present paper is represented by the calculation of two essential harmonic series with a weight 5 structure, involving harmonic numbers of the type H2n . The two main series are evaluated by also exploiting results and strategies presented in the book, (Almost) Impossible Integrals, Sums, and Series, 2019 . Mathematics subject classification (2010): 26A36, 33B30, 65B10, 40C10, 40G10, 11M06.

On the calculation of two essential harmonic series with a weight 5 structure, involving harmonic numbers of the type H_2n