TITLE: Apparent patterns in ratios of consecutive primes
QUESTION [7 upvotes]: I was plotting the values of $\frac{P(n+1)}{P(n)+2}$, where $P(n)$ is the nth prime number. I noticed very easily that the values seem to belong very nicely to a set of "trajectories". They clearly cannot have arbitrarily values.
Of course all these trajectories are decreasing and tend to $1$, except the trajectory for twin primes, which is, of course, $1$.
I also made a rough estimate that they can be estimated with functions of the form $1+a/x^b$, the first trajectory above constant $1$ is close to $1+1.1/x^{1.19}$. 
Is there more info on these trajectories? Given $n$ is smaller than some number, how many trajectories there are in total? By counting from the graph, I would say, if $n<10,000$, there are about 27 trajectories (ie.about 27 different curves where the values fit).

REPLY [5 votes]: The trajectories are simply the curves $1 + \frac{2k}{x \log x} \approx 1 + \frac{2k}{P(\lfloor x\rfloor)}$ for $k = 0, 1, 2, \ldots$.  Since $P(n+1)-P(n)$ tends to be extremely small compared to $n$, it takes the same (even) values quite often, and your function is nothing more than $1 + \frac{P(n+1)-P(n)-2}{P(n)+2}$.