In this paper, we give the exact number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\)-additive cyclic codes of length \(n=r+s,\) for any positive integer r and any positive odd integer s. We will provide a formula for the the number of separable \({{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes of length n and then a formula for the number of non-separable \({{\mathbb {Z}} _{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes of length n. Then, we have generalized our approach to give the exact number of \({{\mathbb {Z}}_{p}{\mathbb { Z}_{p^{2}}}}\)-additive cyclic codes of length \(n=r+s,\) for any prime p, any positive integer r and any positive integer s where \(\gcd \left( p,s\right) =1.\) Moreover, we will provide examples of the number of these codes with different lengths \(n=r+s\).

在本文中，我们给出\（{{\ mathbb {Z}}} _ {2} {{\ mathbb {Z}}} _ {4} \）的确切数-长度为\（n = r + s，\）对于任何正整数r和任何正奇数s。我们将为可分解的\（{{\ mathbb {Z}} _ {2} {{\ mathbb {Z}} _ {4}}} \\） -个长度为n的加和循环码的数量提供一个公式长度为n的不可分\（{{\ mathbb {Z}} _ {2} {{\ mathbb {Z}} _ {4}}} \）的数量的公式。然后，我们对方法进行了一般化，以给出\（{{\ mathbb {Z}} _ {p} {\ mathbb {Z} _ {p ^ {2}}}}} \\）的确切数量长度\（n = r + s，\）对于任何素数p，任何正整数r和任何正整数s，其中\（\ gcd \ left（p，s \ right）= 1。\）此外，我们将提供以下示例这些具有不同长度\（n = r + s \）的代码的数量。