On an Erdős–Straus type problem

Motivated by the Erdős–Straus conjecture, we study whether the Euler totient ratio φ(n)n \frac{\varphi(n)}{n} can be expressed as a sum of three unit fractions, φ(n)n=1x+1y+1z,x,y,z∈ℕ. \frac{\varphi(n)}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}, \quad x, y, z \in \mathbb{N}. We prove that such a representation is impossible for prime powers p^k with p ≥ 11, and for integers of the form p^k1 q^k2 with p ≥ 5 and q ≥ 15p. In contrast, for 2 ≤ p ≤ 7 the representation exists for all p^k. Moreover, for every integer r ≥ 3, there are infinitely many integers n with ω(n)=r for which no such representation exists. Further, if a representation holds, we obtain the bounds 2≤x≤9+3eγlog log n,nxφ(n)x−n≤y≤2nxφ(n)x−n. 2 \le x \le 9 + 3e^{\gamma} \log \log n, \quad \frac{nx}{\varphi(n)x - n} \le y \le \frac{2nx}{\varphi(n)x - n}. In addition, if x≥3(n-1)(eγlog log n+3)n x \geq \frac{3(n-1)(e^{\gamma} \log \log n + 3)}{n} , then rad (n)≤42r-42r-13 (n) \leq \frac{4^{2^{r}} - 4^{2^{r-1}}}{3} , where γ is Euler’s constant and r = ω(n).