Blue Chains

We have been colouring the numbers according to their remainder when divided by 6. Numbers of the form 6n+5, which we call blue numbers, always have a predecessor of the form 4n+3 (which for large n is about 2/3rds of the starting value) eg.

The colour of the this smallest node varies but a third of the time it will also be blue, so we can start constructing a chain of blue numbers, each one roughly 2/3rds of its predecessor. So for 17 in the above list we can construct the larger tree;

Examining a few of these trees we can see that sequences of decreasing blue nodes end in a number of the form 3*2ⁿ-1. The numbers in this sequence start with [5,11,23,47,95,191,...]. If we take a number of this form and put it through the (3x+1)/2 equation n-1 times we get 2*3ⁿ-1

For instance

All the numbers in the above table are of the form 6n+5 and appear blue in the diagram. This shows that we can create an arbitrarily long sequence of increasing values, just by increasing the value of n.

The figure on the left shows the tree generated by these first seven sequences. 

The end number divided by the start (for large n) is 2*3ⁿ/3*2ⁿ, or (1.5)^n-1. Here

which converges on log(3)/log(2), about 1.5849625007211563. If there is a limit to log(Max(N))/log(N), it must be at least this large. This chart shows the largest value of log(Max(N))/log(N) for successive ranges of N of the form (2ⁱ to 2ⁱ⁺¹-1). 

The dotted line is log(3)/log(2). The highlights are

This chart shows the path taken by 270271. The Y axis is Log₁₀.

If we continue we find that 6,631,675 gives a log (Max(N))/log(N) of 1.9502. So, although this blue chains method will find sequences where log (Max(N))/log(N) is about 1.585, there are series with larger values.

The image on the left was drawn using GraphViz, see http://www.research.att.com/sw/tools/graphviz/