A variant of Knapsack problem

You want to minimize $\sum_{i=1}^n c_i a^{x_i}$ subject to $\sum_{i=1}^n a_i x_i \le A$ and $x_i \in \mathbb{Z}^+$.

Define value function $$v(k,B) = \min\left\{\sum_{i=1}^k c_i a^{x_i}: \sum_{i=1}^k a_i x_i \le B \land x_i \in \mathbb{Z}^+\right\}$$ so that the original problem is to calculate $v(n,A)$. The usual dynamic programming approach conditions on the value $y$ of $x_k$ to obtain recursion $$v(k,B) = \min_{y\in \{0,\dots,\lfloor B/a_k \rfloor\}} \left\{c_k a^y + v(k-1,B-a_k y) \right\}.$$

Alternatively, you can linearize the problem by introducing binary decision variables $z_{ij}$ to indicate whether $x_i=j$. The problem is then to minimize the linear objective function $$\sum_{i=1}^n c_i \sum_{j=0}^{\lfloor B/a_i \rfloor} a^j z_{ij}$$ subject to linear constraints \begin{align} \sum_{i=1}^n a_i x_i &\le A \\ \sum_{j=0}^{\lfloor B/a_i \rfloor} j z_{ij} &= x_i &&\text{for $i\in\{1,\dots,n\}$} \\ \sum_{j=0}^{\lfloor B/a_i \rfloor} z_{ij} &= 1 &&\text{for $i\in\{1,\dots,n\}$} \end{align} Now call an integer linear programming solver.