Congruences for Andrews’ (k, i)-singular overpartitions

Andrews recently defined new combinatorial objects which he called (k, i)-singular overpartitions and proved that they are enumerated by C¯k,i(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{C}_{k,i}(n)$$\end{document} which is the number of overpartitions of n in which no part is divisible by k and only the parts ≡±i(modk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv \pm i \pmod {k}$$\end{document} may be overlined. Andrews further showed that C¯3,1(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{C}_{3,1}(n)$$\end{document} satisfies some Ramanujan-type congruences modulo 3. In this paper, we show that for any pair (k, i), C¯k,i(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{C}_{k,i}(n)$$\end{document} satisfies infinitely many Ramanujan-type congruences modulo any power of prime coprime to 6k. We also show that for an infinite family of k, the value C¯3k,k(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{C}_{3k,k}(n)$$\end{document} is almost always even. Finally, we investigate the parity of C¯4k,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{C}_{4k,k}$$\end{document}.