  A Sierpi\'nski number is a positive odd integer $k$ such that $k \cdot 2^n +
1$ is composite for all positive integers $n$. Fix an integer $A$ with $2 \le
A$. We show that there exists a positive odd integer $k$ such that $k\cdot a^n
+ 1$ is composite for all integers $a \in [2, A]$ and all $n \in \mathbb{Z}^+$.
