Interesting pattern in plot involving prime numbers If we plot $f(n) = \dfrac{n+1}{p_{n+1}} - \dfrac{n}{p_n}, n \in \mathbb{N} $, we get an interesting pattern.

Questions:

*

*Why it looks like there's different lines on plot?


*Why they have such shape?


*How those lines can be approximated?


*Why there's like peak at $n \approx 500$?


*Why points in lines become sparcer as they are closer to the bottom?
 A: As Vepir suggested in his comment, the curves correspond to consecutive primes with different prime gaps. The one on the top, for example, correspond to pairs of twin primes. That is $$\frac{n+1}{p_{n+1}}-\frac{n}{p_n} = \frac{n+1}{p_{n}+2}-\frac{n}{p_n}$$
The curves can be approximated using the known approximation for the $n$-th prime. For example, we know that $p_n \approx n\,(\log n + \log\log n -1)$.
Here's how it looks that approximation for the curve on the top.

The curves on the bottom corresponds to pairs of consecutive with larger prime gaps, that naturally occur less frequently.
A: Note that
$$f(n)={1\over p_{n+1}}-n\left({1\over p_n}-{1\over p_{n+1}}\right)={1\over p_{n+1}}-{ng_n\over p_np_{n+1}}\approx{1\over n\ln n+g_n}\left(1-{g_n\over\ln n}\right)$$
where $g_n=p_{n+1}-p_n$ and we've used the crude asymptotic approximation $p_n\approx n\ln n$. So very roughly, we are looking at the family of curves
$$f_k(x)={1\over x\ln x+2k}\left(1-{2k\over\ln x}\right)$$
with $k=1,2,3,\ldots$.  These curves behave qualitatively much like what the OP has observed; their quantitative disagreement -- the curve with $2k=6$ has a peak value of approximately $.00002$ at $x\approx873$, instead of $.00005$ near $500$ -- is due to the crudeness of the approximation $p_n\approx n\ln n$. Using the better approximation mentioned in jjagmath's answer, $p_n\approx n(\ln n+\ln\ln n-1)$, one gets a curve (for $2k=6$) with a peak of approximately $.00004$ at $x\approx431$, which comes closer to the OP's values.
In answer to the OP's final bullet-point question, the sparsity of the lower lines stems, heuristically, from the fact that large prime gaps (i.e., large values of $2k$) take a while to kick in and remain relatively rare among the first $10{,}000$ primes.
