Can recurrences of the form $$ \sum_{i=1}^n a_iX_i=\gcd(n, X_n) $$
Where $a_i$ are constant coeficients. $a_i,X_i$ are integers. $a_n\neq0$.
For $n \geq 2$ be solved?
Here is an example:
$$ X_n=X_{n-1}+\gcd(n, X_{n-1}). X_1=7 $$
This example produces only prime numbers, and recurrences like this arose in some exercises I was working on. I want to know if a closed form is possible.