The series $1,3,6,9,12,14$ represent the first six terms of the series A168045 on OEIS. The formula for this says $a_n = 2n + n / log n + O(n / (log n)^2)$. I do not understand this notation. What base log is this, and how do we compute $O(.)$? Isn't $O(.)$ an order?
Looking a bit more at this, this is derived from some Brun's Theorem.
Alternately, is there an analytical expression for this?
The log is the natural logarithm. $O(.)$ is "Big-O notation" describing the behaviour as $n \to \infty$. That is, the formula says there exist numbers $N$ and $C$ such that $$ \left| a_n - 2 n - \frac{n}{\log n} \right| < C \frac{n}{(\log n)^2} $$ for all $n > N$.