Let $A_n$ be the smallest positive integer satisfying $A_{n}mod{p}_i=i$ for all $1\le i\le n$, where $p_i$ is the $i$-th prime.
For example $A_2=5$, since this is the smallest positive solution of the system of equations

• $A_{2}mod2=1$
• $A_{2}mod3=2$

The system of equations for $A_3$ adds another constraint. That is, $A_3$ is the smallest positive solution of

• $A_{3}mod2=1$
• $A_{3}mod3=2$
• $A_{3}mod5=3$

and hence $A_3=23$. Similarly, one gets $A_4=53$ and $A_5=1523$.

Let $S(n)$ be the sum of all primes up to $n$ that divide at least one element in the sequence $A$.
For example, $S(50)=69=5+23+41$, since $5$ divides $A_2$, $23$ divides $A_3$ and $41$ divides $A_{10}=5765999453$. No other prime number up to $50$ divides an element in $A$.

Find $S(300000)$.