Composite run around primes

We prove that for every positive even integer $2n$, there exist infinitely many sequences of $2n-1$ consecutive composite integers, with each sequence immediately followed or preceded by a prime. Our approach uses factorial and primorial constructions combined with Dirichlet's theorem on arithmetic progressions, yielding explicit formulas that determine the positions of these sequences. This result offers new insights into the distribution of primes relative to consecutive composite numbers.