Question 71
Let $k \geq 3$ an integer. We define the $\left(a_{n}\right)_{n \geq k}$ sequence by $a_{k}=2k$, and $$
a_{n}= \begin{cases}a_{n-1}+1 & \text{ if } \operatorname{pgcd}\left(a_{n-1}, n\right)=1 \\ 2n & \text{ Otherwise. }\end{cases}
$$ show that the $\left(a_{n+1}-a_{n}\right)_{n \geq k}$ sequence has an infinity of terms that are prime numbers.
a_{n}= \begin{cases}a_{n-1}+1 & \text{ if } \operatorname{pgcd}\left(a_{n-1}, n\right)=1 \\ 2n & \text{ Otherwise. }\end{cases}
$$ show that the $\left(a_{n+1}-a_{n}\right)_{n \geq k}$ sequence has an infinity of terms that are prime numbers.