erdos_321.variants.upper

Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that $$ R(N) \le \frac{1}{\log 2} \log_r N \left( \frac{N}{\log N} \prod_{i=3}^{r} \log_i N \right), $$ valid for any $k \ge 4$ with $\log_k N \ge k$ and any $r \ge 1$ with $\log_{2r} N \ge 1$. (In these bounds $\log_i n$ denotes the $i$-fold iterated logarithm.) [BlEr75] Bleicher, M. N. and Erdős, P., _The number of distinct subsums of $\sum \sb{1}\spN\,1/i$_. Math. Comp. (1975), 29-42. [BlEr76b] Bleicher, Michael N. and Erdős, Paul, _Denominators of Egyptian fractions. II_. Illinois J. Math. (1976), 598-613.

theorem erdos_321.variants.upper (N r : ℕ) (hr : 1 ≤ r) (hrN : 1 ≤ log^[2 * r] N) :
    R N ≤ 1 / log 2 * log^[r] N * N / log N * ∏ i ∈ Finset.Icc 3 r, (log^[i] N)

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