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References
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The series\(\sum {\frac{1}{{\alpha _v }}}\) is ussually called Engel-series of second kind; for its properties and for a proof of (I. 2)–(I. 3) see e. g.P. Erdős, On the integer solutions of the equation\(\frac{1}{{x_1 }} + ... + \frac{1}{{x_n }} = \frac{a}{b}\) (in Hungarian),Matematikai Lapok,I.3 (1950), pp. 192–210.
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Erdős, P., Turán, P. On some problems of a statistical group-theory. IV. Acta Mathematica Academiae Scientiarum Hungaricae 19, 413–435 (1968). https://doi.org/10.1007/BF01894517
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DOI: https://doi.org/10.1007/BF01894517