In this paper, we present a study on the number of \(\mathbb {Z}_2\mathbb {Z}_4\mathbb {Z}_8\) -additive cyclic codes of length \(\gamma +\delta +\omega\) , where \(\gamma\) is any positive integer and \(\delta\) and \(\omega\) are odd positive integers. After exploring their algebraic structure, we give a formula for the number of separable \(\mathbb {Z}_2\mathbb {Z}_4\mathbb {Z}_8\) -additive cyclic codes, and we also discuss a formula for the number of their non-separable codes in different cases of their generator polynomials. Along the similar line, we discuss both cases of non-separable cyclic codes that either lie in the ring \(\frac{\mathbb {Z}_2[x]}{\langle x^\gamma -1\rangle }\times \frac{\mathbb {Z}_4[x]}{\langle x^\delta -1\rangle }\times \frac{\mathbb {Z}_8[x]}{\langle x^\omega -1\rangle }\) with \(\gcd (\gamma , \delta ,\omega )=1\) or \(\gcd (\gamma , \delta ,\omega )\ne 1\) as the case may be. Further, we generalize our study to the number of \(\mathbb {Z}_p\mathbb {Z}_{p^2}\mathbb {Z}_{p^3}\) -additive cyclic codes of length \(\gamma +\delta +\omega\) , for positive integers \(\gamma ,\delta ,\omega\) and prime p such that \(\gcd (\delta ,p)=1\) and \(\gcd (\omega ,p)=1\) . Moreover, we enumerate the number of this family of codes of various lengths \(\gamma +\delta +\omega\) to illustrate our results.
