Hi,
Using these definitions:
Prime pair of n definition: For a given number n and for two primes x and y, with x between n-n and n and y between n and n+n, if n-x is equal to y-n then x and y are called a prime pair z(x y) of n.
Primal pair: For prime pair z(x y) if n-x is a prime number then prime pair z is called a primal pair Z(x y). It follows if n-x is a prime number then y-n is also a prime number as y-n = n-x
Primorial numbers n, ie 6, 30, 210, 2310, 30030 and to a lesser extent multiples of primorials ie 30,60,90,..240, etc have more primal pair's Z associated with them than any other numbers.
The count of primal pair's Z increases with larger primorial numbers n.
The distribution of the primal pair's Z for a given n occurs in the range n-n to n+n, in other words every number n can have primal pairs in the range from zero to n+n, which is why for larger numbers the count of primal pair's Z increases.
I created this graph with numbers 0 to 8167 on the x-axis, and the y-axis is the Z(count) for each number.
As can be seen there is a large gap between the numbers that have a Z(count) = 0, and the other numbers (primorial multiples) that have Z(count) greater than zero.
What causes this big gap? It is a bit strange as the gap being there basically is saying there is a whole range of Z(counts) for any n that can't occur.
cheers, Jamie