Let $a_1=2$ and for all natural number $n$, define $a_{n+1}=a_n(a_n+1)$. Then, as $n→∞$, the number of prime factors of $a_n$ are increasing indefinitely, increasing to a finite limit, decreasing, or oscillating?
The answer is not given to it in the worksheet I refer to, so I wanted to confirm my reasoning and answer.
My Reasoning: We have $a_{n+1}=a_n(a_n+1)$, which means that all the factors of $a_n$ are also the factors of $a_{n+1}$. Also we know that $\gcd(a_n,a_n+1)=1$, which means that they share no common prime factors, hence $a_n+1$ must be contributing some other prime factors to $a_{n+1}$ which $a_n$ didn't contribute. This means that the prime factors for $a_{n+1}$ are increasing indefinitely (going to infinity) as $n\rightarrow \infty$
Please check whether the reasoning is correct, and if not, please provide another approach.
THANKS