Topological properties and algebraic independence of sets of prime-representing constants

Let (Formula presented.) be a sequence of positive integers. We investigate the set of (Formula presented.) such that the integer part of (Formula presented.) is always a prime number for every positive integer k. Let (Formula presented.) be this set. The first goal of this article is to determine the topological structure of (Formula presented.). Under some conditions on (Formula presented.), we reveal that (Formula presented.) is homeomorphic to the Cantor middle third set for some a. The second goal is to propose an algebraically independent subset of (Formula presented.) if (Formula presented.) is rapidly increasing. As a corollary, we disclose that the minimum of (Formula presented.) is transcendental. In addition, we apply the main result to (Formula presented.) in the case when (Formula presented.). As a consequence, we give an algebraically independent and countably infinite subset of this set. Furthermore, we also get results on the rational approximation, (Formula presented.) -linear independence, and numerical calculations of elements in (Formula presented.).