Arithmetic properties of overpartition triples

Let p¯3(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline p _3}\left( n \right)$$\end{document} be the number of overpartition triples of n. By elementary series manipulations, we establish some congruences for p¯3(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline p _3}\left( n \right)$$\end{document} modulo small powers of 2, such as p¯3(16n+14)≡0(mod32),p¯3(8n+7)≡0(mod64)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline p _3}\left( {16n + 14} \right) \equiv 0\left( {\bmod 32} \right),{\overline p _3}\left( {8n + 7} \right) \equiv 0\left( {\bmod 64} \right)$$\end{document}. We also find many arithmetic properties for p¯3(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline p _3}\left( n \right)$$\end{document} modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers α ≥ 1 and n ≥ 0, we p¯3(32α+1(3n+2))≡0(mod9·24),p¯3(42α−1(56n+49))≡0(mod7),p¯3(72α+1(7n+3))≡p¯3(72α+1(7n+5))≡p¯3(72α+1(7n+6))≡0(mod7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar p_3 \left( {3^{2\alpha + 1} \left( {3n + 2} \right)} \right) \equiv 0\left( {\bmod 9\cdot2^4 } \right),\bar p_3 \left( {4^{2\alpha - 1} \left( {56n + 49} \right)} \right) \equiv 0\left( {\bmod 7} \right),\bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 3} \right)} \right) \equiv \bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 5} \right)} \right) \equiv \bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 6} \right)} \right) \equiv 0\left( {\bmod 7} \right)$$\end{document}, and for r ∈ {1, 2, 3, 4, 5, 6}, p¯3(11⋅74α−1(7n+r))≡0(mod11)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline p _3}\left( {11 \cdot {7^{4\alpha - 1}}\left( {7n + r} \right)} \right) \equiv 0\left( {\bmod 11} \right)$$\end{document}.