Let \({\overline{A}}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In this paper, we prove that \({\overline{A}}_{\ell }(n)\) is almost always divisible by \(p_i^j\) if \(p_i^{2a_i}\ge \ell \), where j is a fixed positive integer and \(\ell =p_1^{a_1}p_2^{a_2} \dots p_m^{a_m}\) with primes \(p_i>3\). We obtain a Ramanujan-type congruence for \({\overline{A}}_{7}\) modulo 7. We also exhibit infinite families of congruences and multiplicative identities for \({\overline{A}}_{5}(n)\).

令\（{\ overline {A}} _ {\ ell}（n）\）是n分割成不可被\（\ ell \）整除的部分的数量。在本文中，我们证明\（{\ overline {A}} _ {\ ell}（n）\）几乎总是可以由\（p_i ^ j \）整除，如果\（p_i ^ {2a_i} \ ge \ ell \），其中j是一个固定的正整数，而\（\ ell = p_1 ^ {a_1} p_2 ^ {a_2} \ dots p_m ^ {a_m} \）带有质数\（p_i> 3 \）。我们获得模（7）\（{\ overline {A}} _ {7} \）的Ramanujan型同余。我们还展示了\（{\ overline {A}} _ {5}的无穷大同余和乘法恒等式。（n）\）。