The largest integer expressible as a sum of reciprocal of integers
Given \(n\), denote by \(N(n)\) the set of integers \(k\), representable as a sum of at most \(n\) Egyptian fractions with denominators \(\leq n\), i. e. in the form \[ k = \sum_{\nu = 1}^n {\varepsilon_\nu \over \nu}, \quad \text{ where } \varepsilon_\nu \in \{0,1\}. \] The largest integer in this set is denoted by \( \displaystyle M(n) = \max_{a \in N(n)} a. \) The author is interested in precise estimates of \( M(n) \). He shows: There exist constants \( c_1, c_2, n \), such that for all \( n > n_0 \) the estimate \[ \log n + \gamma - 2 - {c_1 \over \log \log n} \leq M(n) \leq \log n + \gamma - {c_2 \over \log \log n} \] holds. \(\gamma\) is Euler's constant. For the complicated proof of this result in combinatorial number theory, the author makes use of Lemmata of several of his former papers [J. Number Theory 28, 272-282 (1988; Zbl 0637.10007); Can. Math. Bull. 33, 235-241 (1990; Zbl 0656.10012)].
precise estimate of the largest integer representable as sum of reciprocals of different integers
scientific article; zbMATH DE number 1309361