  Let $(c_k)_{k\in \mathbb{N}}$ be a sequence of positive integers. We
investigate the set of $A>1$ such that the integer part of $A^{c_1\cdots c_k}$
is always a prime number for every positive integer $k$. Let $\mathcal{W}(c_k)$
be this set. The first goal of this article is to determine the topological
structure of $\mathcal{W}(c_k)$. Under some conditions on $(c_k)_{k\in
\mathbb{N}}$, we reveal that $\mathcal{W}(c_k)\cap [0,a]$ is homeomorphic to
the Cantor middle third set for some $a$. The second goal is to propose an
algebraically independent subset of $\mathcal{W}(c_k)$ if $c_k$ is rapidly
increasing. As a corollary, we disclose that the minimum of $\mathcal{W}(k)$ is
transcendental. In addition, we apply the main result to the set of $A>1$ such
that the integer part of $A^{3^{k!}}$ is always a prime number. As a
consequence, we give a certain infinite subset of this set which is
algebraically independent. Furthermore, we also get results on the rational
approximation, $\mathbb{Q}$-linear independence, and numerical calculations of
elements in $\mathcal{W}(c_k)$.
