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) begining 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 assymetric 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