Ternary Egyptian fractions with prime denominator

For a prime number p, let A3(p)=|{m∈N:∃m1,m2,m3∈N,mp=1m1+1m2+1m3}|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_3(p)= | \{ m \in \mathbb {N}: \exists m_1,m_2,m_3 \in \mathbb {N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \} |$$\end{document}. In 2019 Luca and Pappalardi proved that x(logx)3≪∑p≤xA3(p)≪x(logx)5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x (\log x)^3 \ll \sum _{p \le x} A_{3}(p) \ll x (\log x)^5$$\end{document}. We improve the upper bound, showing ∑p≤xA3(p)≪x(logx)3(loglogx)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{p \le x} A_{3}(p) \ll x (\log x)^3 (\log \log x)^2$$\end{document}.