Let's consider the following metric of the gap between consecutive primes $$m(k)=\frac {p_k^2-p_{k-1}^2} {24}\;\;\;\;\;(k\ge4)$$

Now, let's define the function

$\delta(k)=m(k)\;\;\;\;$ if $\,m(k)\,$ is prime $\;\;\;\;(1)$

$\delta(k)=0\;\;\;\;\;\;\;\;\;\;$ otherwise $\,\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$

If we plot the graph of $\,m(k)\,$ we obtain (up to $\,k=10^4$):

But, if we plot the graph of $\,\delta(k)$, we obtain **something completely different** (up to $\,k=10^4$):

In the second graph the three different increasing curves seem to have the following asymptotic behaviors:

$$\sim \frac {p_k} {2}$$

$$\sim \frac {p_k} {3}$$

$$\sim \frac {p_k} {6}$$

Further, the prime numbers are not uniformly distributed among the three curves. The number of primes decreases passing from the highest to the lowest curve(I'm trying to estimate the number of primes that fall on each curve).I do not have adequate knowledge of analytic number theory to explain the phenomenon, therefore I ask for your help.

Many thanks.