Generalization of the Collatz Conjecture via Dynamic Primorial Multipliers and Divisors

We propose a generalization of the Collatz conjecture based on the fundamental properties of prime numbers. While the original $3x+1$ problem restrictedly employs the prime $2$ as a divisor and $3$ as a multiplier, this paper introduces a recursive mapping system. We define a map where an integer is divided by the set of the first $n$ primes and multiplied by the $(n+1)$-th prime. We conjecture that for any $n$, the sequence always converges to a trivial cycle containing 1.