Direction of Goldbach Partitions

[Paul Mackenzie]

I have been investigating goldbach partitions for some time.
One interesting observation I have been able to determine is concerning the "direction" of the goldbach partitions whether they are increasing or decreasing as 2N increases.

To get an idea of this I constructed a function f(2N) where

f(2N) = 1 if G[2N] - G[2N-2] > 0
f(2N) = 0 if G[2N] - G[2N-2] = 0
f(2N) = -1 if G[2N] - G[2N-2] < 0


where G[2N] is the number of partitions for the even number 2N.

I then generated a sequence of numbers f(2N) beginning at 2N = 10 and finishing at 2N=75398
[arbitrarily selected]. The sequence is then:

{1,1,-1,1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1,1,1,-1,1,1,-1,-1,1,-1,1,1,-1,1,1,-1,...}


I ran a program to determine a histogram of the number of occurences of the subsequences of the following form


histogram

subsequence type
No. of Occurences

x,1,x
5563

x,-1,x
7003

x,1,1,x
7003

x,-1,-1,x
5562

x,1,1,1,x
0

x,-1,-1,-1,x
1

x,0,x
0





I have been trying to work out a descriptive name for this sequence. Any suggestions?

The question then arises will a sequence generated from Hardy-Littlewoods equation
for Goldbach partions be similar?

Also , any comments on the histogram? Why the asymmetric nature ?

And why only these subsequences? I would have thought there would have been longer runs. Any ideas?

Paul

 

[haruspex]

What exactly does 'x,1,x' mean? Are the two x's the same or independent? Either way, the count seems much too low. How can it be less than x,1,1,x?

 

[Paul Mackenzie]

Hi Haruspex:

I am counting the number of consecutive runs of similar digits in the sequence. So the run x1x refers to either the subsequence 0,1,0, or 0,1,-1 or -1,1,0 or -1,1,-1.
As another example the consecutive run of two positive ones viz x,1,1,x refers to the subsequences -1,1,1,0 or -1,1,1,-1 or 0,1,1,0 or 0,1,1,-1.

All the counts are mutually exclusive, and add up to the correct number of digits [give or take one or two, as I am having problems with the start and end of the sequence]

But the question remains why this particular pattern.

Kind Regards
Paul