User:Jon Awbrey/Riffs and Rotes
Author: Jon Awbrey
Contents
Idea
Let be the prime, where the positive integer is called the index of the prime and the indices are taken in such a way that Thus the sequence of primes begins as follows:

The prime factorization of a positive integer can be written in the following form:
where is the prime power in the factorization and is the number of distinct prime factors dividing The factorization of is defined as in accord with the convention that an empty product is equal to
Let be the set of indices of primes that divide and let be the number of times that divides Then the prime factorization of can be written in the following alternative form:
For example:

Each index and exponent appearing in the prime factorization of a positive integer is itself a positive integer, and thus has a prime factorization of its own.
Continuing with the same example, the index has the factorization and the index has the factorization Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:

This leads to the following development:

Continuing to replace every index and exponent with its factorization produces the following development:

The 's that appear as indices and exponents are formally redundant, conveying no information apart from the places they occupy in the resulting syntactic structure. Leaving them tacit produces the following expression:

The pattern of indices and exponents illustrated here is called a doubly recursive factorization, or DRF. Applying the same procedure to any positive integer produces an expression called the DRF of If is the set of positive integers, is the set of DRF expressions, and the mapping defined by the factorization process is denoted then the doubly recursive factorization of is denoted
The forms of DRF expressions can be mapped into either one of two classes of graphtheoretical structures, called riffs and rotes, respectively.
is the following digraph: 
is the following graph: 
Riffs in Numerical Order




























































Rotes in Numerical Order




























































Prime Animations
Riffs 1 to 60
Rotes 1 to 60
Selected Sequences
A061396
 Number of "rooted indexfunctional forests" (Riffs) on n nodes.
 Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.
 
 
 
 

A062504
 Triangle in which kth row lists natural number values for the collection of riffs with k nodes.

 
 
 
 
 

A062537
 Nodes in riff (rooted indexfunctional forest) for n.




























































A062860
 Smallest j with n nodes in its riff (rooted indexfunctional forest).










A109301
 a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.
 Example




























































Miscellaneous Examples
 
