Riffs & Rotes & A061396 & A062504?

[Marc LeBrun]

It might be worth noting that these trees themselves can be encoded as 
numbers, thus forming a kind of Godelization of the integers.

Let me sketch an example for one of the notations I think I saw go by here 
recently:

A given number corresponds to an infinite sequence of finite trees, all but 
a finite number of which are empty.  We wish to transcribe these into an 
infinite string of bits, all but a finite number of which are 0.

First we assign each tree (ie each prime) its own infinite subset of 
bits.  To the tree rooted at 2 we assign the even bit positions, to the 
tree rooted at 3 we assign every other of the remaining positions (ie the 
bit positions congruent to 1 mod 4), to the tree rooted at 5 we interleave 
again (ie 3 mod 8) and so on, sort of giving each successive prime "half" 
the remaining positions.

Next we transcribe the tree rooted at a given prime into these 
positions.  If the tree is empty, o, we set all the bits to 0.  If the tree 
is one, (o), we set the least significant bit to a 1 and the rest to 0.

For the more complex trees we just recurse (I think the exact details of 
the recursion you pick gives you a choice of the different structures 
discussed in this thread, but I'm not sure I understand all that yet!).

Anyway, the point is that this allows you to map the integers through 
the  trees and back into numbers, which opens up many new possibilities.

We needn't be limited to just taking censi of various populations (eg trees 
of various types).  We can also directly access the images of single 
specific concrete cases numerically.

For example if f is the mapping (either as we've constructed above, or some 
other way) then there's sequences such as f(n+1) and tables of operations 
such as f(x+y) or f(xy), and for all of these there's also their images 
under the inverse of f.  All of which might be entered in the EIS, I imagine.

And so on...