Why is there an unexpected increase in the density of certain types of Goldbach primes? Note: Posted in MO since it was unanswered in MSE.
I was checking how quickly we can verify Goldbach's conjecture for a given even number $n$ and it was clear that searching backward starting from the largest prime below $n$ was much faster than a forward search starting from the small odd prime $3$. I observed the following.
Consider the set of even numbers $n$ which can be written as the sum of two primes and $p$ be the largest prime less than $n$ such that $n-p$ is also a prime. Let $p_k$ be the $k$-th prime. Then, $p_{\pi(n)-2}$, $p_{\pi(n)-1}$ and $p_{\pi(n)}$ are the three largest primes preceding $n$. Now $p$ can either be the largest prime below $n$, or the prime just before the largest prime or the prime before that and so on. Accordingly, we define:

*

*$d_1$ is the density of $n$ such that $p = p_{\pi(n)}$

*$d_2$ is the density of $n$ such that $p = p_{\pi(n)-1}$

*$d_3$ is the density of $n$ such that $p = p_{\pi(n)-2}$
Data for $n \le 3 \times 10^{10}$ shows that $d_1$ and $d_2$ decreased as $n$ increased. This was intuitively expected as explained in the answer for the related question "If $p$ is the largest prime less than $2n$, what is the probability that $2n-p$ is a prime?". However, quite counter intuitively, $d_3$ increased with $n$ as shown in the graphs below.
Question 1: Why does $d_3$ increase with $n$ while $d_1$ and $d_2$ decreased?
Question 2: What is the limiting value of $d_3$? Since $d_3 < 1$ and is increasing, it must converge to a positive limiting value.
Note: I have not computed $d_k$ for $k \ge 4$ so I do not know if this increasing trend is observed other values of $k$.
Graphs in normal scale



 A: All $d_k$ go to zero, simply because the density of primes goes to zero. Let $\epsilon > 0$ and pick $N$ such that the density of primes in any interval of length at least $N$ is less than $\epsilon$. Since the density of primes goes to zero, almost all even integers lie in a prime gap of size larger than $N$, say between $p_a$ and $p_{a+1}$. The even numbers $n$ between $p_a$ and $p_{a+1}$ 'belong' to $d_k$ if $n-p_{a+1-k}$ is prime. The numbers $n-p_{a+1-k}$ lie in an interval of length at least $N$, so the density of primes here is less than $\epsilon$, showing that $d_k<\epsilon$.
The reason that $d_3$ increases at first, is because al lot of numbers were taken from $d_1$ and $d_2$ before. I suspect that if you plot the density of integers $n$ such that $n-p_{\pi(n)-2}$ is prime, it will start decreasing much sooner.
A: Disclaimer: this is not a real answer but rather a heuristics that is too long for a comment.
Building upon my comment, denote by $k_{0}(m):=\pi(m+r_{0}(m))-\pi(m-r_{0}(m))$ and say $m$ is $K$-central if $k_{0}(m)=K$. Under some reasonable hypotheses, the number of $K$-central numbers not exceeding $x$ is asymptotically $\frac{\pi(x)}{K}$ for positive $K$ (edit: see this question of mine: Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?).
Your observation suggests one has $k_{0}(n/2)/2\approx 3$ hence $k_{0}(n/2)\approx 6$. As you count half the primes below $n$ summing to $n$, namely the smallest summands, one can expect a density of $\frac{1}{2}\frac{\pi(n)}{6\pi(n)}=1/12$ so I expect the limit of $d_{3}$, if it exists, to equal $1/12$.
Edit: as for the increasing behavior, I came up with a somewhat hand-wavy idea to explain it, so take it with a grain of salt.
According to the jumping champions conjecture, the most frequent prime gap below some $10^{35}$ is $6$. As $k_{0}(m)$ counts the number of prime gaps in $[m-r_{0}(m),m+r_{0}(m)]$, one can expect $2r_{0}(m)$ to be $\asymp 6k_{0}(m)$ in this range. On the other hand, the Hardy-Littlewood $k$-tuples conjecture predicts the smallest diameter of an admissible $7$-tuple, corresponding to a chain of $7-1=6$ prime gaps to equal $20$, while my formula in the link I gave predicts a value of $6(1+H_{6})\approx 20.7\approx \gamma 6^{2}$ where $\gamma$ is the Euler-Mascheroni constant. As the jumping champions are conjecturally $4$ and the primorials, this suggest $d_{3}$ will eventually decrease after some (huge) threshold and that the value $1/12$ is not its limit but its maximum.
