In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function, Andrews defined the so-called singular overpartitions. Singular overpartition function ${\bar{C}}_{k,i}(n)$ 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. Andrews also proved two beautiful Ramanujan type congruences modulo 3 satisfied by ${\bar{C}}_{3,1}(n)$. Later on, Aricheta proved that for an infinite family of ℓ, ${\bar{C}}_{3\ell ,\ell }(n)$ is almost always divisible by 2. In this article, for an infinite subfamily of ℓ considered by Aricheta, we prove that ${\bar{C}}_{3\ell ,\ell }(n)$ is almost always divisible by arbitrary powers of 2. We also prove that ${\bar{C}}_{3\ell ,\ell }(n)$ is almost always divisible by arbitrary powers of 3 when $\ell =3,6,12,24$. Proofs of our density results rely on the modularity of certain eta-quotients which arise naturally as generating functions for the Andrews' singular overpartition functions.

为了给出普通配分函数的 Rogers-Ramanujan 型定理的过配分类似物，安德鲁斯定义了所谓的奇异过配分。奇异过分函数${\bar{C}}_{克,\mathrm{一世}}(n)$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。安德鲁斯还证明了两个美丽的拉马努金型同余满足模 3${\bar{C}}_{3,1}(n)$. 后来，Aricheta 证明了对于一个无穷大的ℓ族，${\bar{C}}_{3\ell ,\ell }(n)$几乎总能被 2 整除。 在本文中，对于Aricheta 考虑的ℓ的无限子族，我们证明${\bar{C}}_{3\ell ,\ell }(n)$ 几乎总是可以被 2 的任意幂整除。 我们还证明 ${\bar{C}}_{3\ell ,\ell }(n)$ 几乎总能被 3 的任意幂整除，当 $\ell =3,6,12,24$. 我们的密度结果的证明依赖于某些 eta 商的模块化，这些 eta 商是作为安德鲁斯奇异过分配函数的生成函数而自然出现的。