The prime-increment conjecture for the LCM recurrence

Let $(a_n)$ be defined by

$$a_1=1,$$

and

$$a_n=a_{n-1}+\operatorname{lcm}(n,a_{n-1})\qquad(n\geq 2).$$

Define the multiplicative increments

$$b_n=\frac{a_n}{a_{n-1}}-1=\frac{n}{\operatorname{gcd}(n,a_{n-1})}.$$

Prime-increment conjecture. For every $n\geq 2$, we have $b_n\in\{1\}\cup\mathbb{P}$, where $\mathbb{P}$ denotes the set of prime numbers.

Numerical evidence supports the claim, but a proof for every $n$ remains open. The paper proves unconditionally that the claim holds for a set of integers of asymptotic density $1$, using a Companion--Sieve framework, the Bombieri--Vinogradov theorem, and elementary sieve bounds.