Automorphisms of $Z_{p^{i_1}}*Z_{p^{i_2}}*...*Z_{p^{i_n}}$

Question

If $Z_{p^{i_1}}\times Z_{p^{i_2}}\times\cdots\times Z_{p^{i_n}}=\langle a_1,...,a_n\rangle$, then each automorphism of this group is the forms as follows,

$$\sigma:a_j\rightarrow a_1^{\ell_{j1}}\cdots a_n^{\ell_{jn}}.$$

Then how to give the necessary and sufficient condition of every $\ell_{jk}$? Or can you give me some references about this question? Thank you very much!