Andrews introduced the singular overpartition function ${\bar{C}}_{k,i}(n)$ which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i(\mathrm{mod}k)$ may be overlined. In this article, we study the divisibility properties of ${\bar{C}}_{4k,k}(n)$ and ${\bar{C}}_{6k,k}(n)$ by arbitrary powers of 2 and 3 for infinite families of k. For an infinite family of k, we prove that ${\bar{C}}_{4k,k}(n)$ is almost always divisible by arbitrary powers of 2. We also prove that ${\bar{C}}_{6k,k}(n)$ is almost always divisible by arbitrary powers of 3 for an infinite family of k. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of 2 satisfied by ${\bar{C}}_{4\cdot 2^{\alpha },2^{\alpha }}(n)$ and ${\bar{C}}_{4\cdot 3\cdot 2^{\alpha },3\cdot 2^{\alpha }}(n)$.

Andrews 介绍了奇异的超分函数 ${\bar{C}}_{克,\mathrm{一世}}(n)$它计算n的过度划分的数量，其中没有部分可以被k整除，只有部分$\equiv \pm \mathrm{一世}(\mathrm{模组}克)$可能会被划线。在本文中，我们研究了可分性${\bar{C}}_{4克,克}(n)$ 和 ${\bar{C}}_{6克,克}(n)$对于k 的无限族，通过 2 和 3 的任意幂。对于k的无限族，我们证明${\bar{C}}_{4克,克}(n)$ 几乎总是可以被 2 的任意幂整除。 我们还证明 ${\bar{C}}_{6克,克}(n)$对于k的无限族，几乎总是可以被 3 的任意幂整除。使用 Ono 和 Taguchi 对 Hecke 算子的幂零性的结果，我们找到了无限的同余族，模 2 的任意幂满足以下条件${\bar{C}}_{4\cdot 2^{\alpha },2^{\alpha }}(n)$ 和 ${\bar{C}}_{4\cdot 3\cdot 2^{\alpha },3\cdot 2^{\alpha }}(n)$.