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Riffs and Rotes

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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 graph-theoretical structures, called riffs and rotes, respectively.

is the following digraph:
Riff 123456789 Big.jpg
is the following graph:
Rote 123456789 Big.jpg

Riffs in Numerical Order

 



Riff 2 Big.jpg



Riff 3 Big.jpg



Riff 4 Big.jpg



Riff 5 Big.jpg



Riff 6 Big.jpg



Riff 7 Big.jpg



Riff 8 Big.jpg



Riff 9 Big.jpg



Riff 10 Big.jpg



Riff 11 Big.jpg



Riff 12 Big.jpg



Riff 13 Big.jpg



Riff 14 Big.jpg



Riff 15 Big.jpg



Riff 16 Big.jpg



Riff 17 Big.jpg



Riff 18 Big.jpg



Riff 19 Big.jpg



Riff 20 Big.jpg



Riff 21 Big.jpg



Riff 22 Big.jpg



Riff 23 Big.jpg



Riff 24 Big.jpg



Riff 25 Big.jpg



Riff 26 Big.jpg



Riff 27 Big.jpg



Riff 28 Big.jpg



Riff 29 Big.jpg



Riff 30 Big.jpg



Riff 31 Big.jpg



Riff 32 Big.jpg



Riff 33 Big.jpg



Riff 34 Big.jpg



Riff 35 Big.jpg



Riff 36 Big.jpg



Riff 37 Big.jpg



Riff 38 Big.jpg



Riff 39 Big.jpg



Riff 40 Big.jpg



Riff 41 Big.jpg



Riff 42 Big.jpg



Riff 43 Big.jpg



Riff 44 Big.jpg



Riff 45 Big.jpg



Riff 46 Big.jpg



Riff 47 Big.jpg



Riff 48 Big.jpg



Riff 49 Big.jpg



Riff 50 Big.jpg



Riff 51 Big.jpg



Riff 52 Big.jpg



Riff 53 Big.jpg



Riff 54 Big.jpg



Riff 55 Big.jpg



Riff 56 Big.jpg



Riff 57 Big.jpg



Riff 58 Big.jpg



Riff 59 Big.jpg



Riff 60 Big.jpg



Rotes in Numerical Order

Rote 1 Big.jpg



Rote 2 Big.jpg



Rote 3 Big.jpg



Rote 4 Big.jpg



Rote 5 Big.jpg



Rote 6 Big.jpg



Rote 7 Big.jpg



Rote 8 Big.jpg



Rote 9 Big.jpg



Rote 10 Big.jpg



Rote 11 Big.jpg



Rote 12 Big.jpg



Rote 13 Big.jpg



Rote 14 Big.jpg



Rote 15 Big.jpg



Rote 16 Big.jpg



Rote 17 Big.jpg



Rote 18 Big.jpg



Rote 19 Big.jpg



Rote 20 Big.jpg



Rote 21 Big.jpg



Rote 22 Big.jpg



Rote 23 Big.jpg



Rote 24 Big.jpg



Rote 25 Big.jpg



Rote 26 Big.jpg



Rote 27 Big.jpg



Rote 28 Big.jpg



Rote 29 Big.jpg



Rote 30 Big.jpg



Rote 31 Big.jpg



Rote 32 Big.jpg



Rote 33 Big.jpg



Rote 34 Big.jpg



Rote 35 Big.jpg



Rote 36 Big.jpg



Rote 37 Big.jpg



Rote 38 Big.jpg



Rote 39 Big.jpg



Rote 40 Big.jpg



Rote 41 Big.jpg



Rote 42 Big.jpg



Rote 43 Big.jpg



Rote 44 Big.jpg



Rote 45 Big.jpg



Rote 46 Big.jpg



Rote 47 Big.jpg



Rote 48 Big.jpg



Rote 49 Big.jpg



Rote 50 Big.jpg



Rote 51 Big.jpg



Rote 52 Big.jpg



Rote 53 Big.jpg



Rote 54 Big.jpg



Rote 55 Big.jpg



Rote 56 Big.jpg



Rote 57 Big.jpg



Rote 58 Big.jpg



Rote 59 Big.jpg



Rote 60 Big.jpg



Prime Animations

Riffs 1 to 60

Animation Riff 60 x 0.16.gif

Rotes 1 to 60

Animation Rote 60 x 0.16.gif

Selected Sequences

A061396

  • Number of "rooted index-functional forests" (Riffs) on n nodes.
  • Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.
    Rote 1 Big.jpg  
Riff 2 Big.jpg Rote 2 Big.jpg

Riff 3 Big.jpg Rote 3 Big.jpg

Riff 4 Big.jpg Rote 4 Big.jpg

Riff 5 Big.jpg Rote 5 Big.jpg

Riff 6 Big.jpg Rote 6 Big.jpg

Riff 7 Big.jpg Rote 7 Big.jpg

Riff 8 Big.jpg Rote 8 Big.jpg

Riff 9 Big.jpg Rote 9 Big.jpg

Riff 16 Big.jpg Rote 16 Big.jpg

A062504

  • Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.

    Rote 1 Big.jpg
Riff 2 Big.jpg Rote 2 Big.jpg

Riff 3 Big.jpg Rote 3 Big.jpg

Riff 4 Big.jpg Rote 4 Big.jpg

Riff 5 Big.jpg Rote 5 Big.jpg

Riff 6 Big.jpg Rote 6 Big.jpg

Riff 7 Big.jpg Rote 7 Big.jpg

Riff 8 Big.jpg Rote 8 Big.jpg

Riff 9 Big.jpg Rote 9 Big.jpg

Riff 16 Big.jpg Rote 16 Big.jpg

Riff 10 Big.jpg Rote 10 Big.jpg

Riff 11 Big.jpg Rote 11 Big.jpg

Riff 12 Big.jpg Rote 12 Big.jpg

Riff 13 Big.jpg Rote 13 Big.jpg

Riff 14 Big.jpg Rote 14 Big.jpg

Riff 17 Big.jpg Rote 17 Big.jpg

Riff 18 Big.jpg Rote 18 Big.jpg

Riff 19 Big.jpg Rote 19 Big.jpg

Riff 23 Big.jpg Rote 23 Big.jpg

Riff 25 Big.jpg Rote 25 Big.jpg

Riff 27 Big.jpg Rote 27 Big.jpg

Riff 32 Big.jpg Rote 32 Big.jpg

Riff 49 Big.jpg Rote 49 Big.jpg

Riff 53 Big.jpg Rote 53 Big.jpg

Riff 64 Big.jpg Rote 64 Big.jpg

Riff 81 Big.jpg Rote 81 Big.jpg

Riff 128 Big.jpg Rote 128 Big.jpg

Riff 256 Big.jpg Rote 256 Big.jpg

Riff 512 Big.jpg Rote 512 Big.jpg

Riff 65536 Big.jpg Rote 65536 Big.jpg

A062537

  • Nodes in riff (rooted index-functional forest) for n.

 



Riff 2 Big.jpg



Riff 3 Big.jpg



Riff 4 Big.jpg



Riff 5 Big.jpg



Riff 6 Big.jpg



Riff 7 Big.jpg



Riff 8 Big.jpg



Riff 9 Big.jpg



Riff 10 Big.jpg



Riff 11 Big.jpg



Riff 12 Big.jpg



Riff 13 Big.jpg



Riff 14 Big.jpg



Riff 15 Big.jpg



Riff 16 Big.jpg



Riff 17 Big.jpg



Riff 18 Big.jpg



Riff 19 Big.jpg



Riff 20 Big.jpg



Riff 21 Big.jpg



Riff 22 Big.jpg



Riff 23 Big.jpg



Riff 24 Big.jpg



Riff 25 Big.jpg



Riff 26 Big.jpg



Riff 27 Big.jpg



Riff 28 Big.jpg



Riff 29 Big.jpg



Riff 30 Big.jpg



Riff 31 Big.jpg



Riff 32 Big.jpg



Riff 33 Big.jpg



Riff 34 Big.jpg



Riff 35 Big.jpg



Riff 36 Big.jpg



Riff 37 Big.jpg



Riff 38 Big.jpg



Riff 39 Big.jpg



Riff 40 Big.jpg



Riff 41 Big.jpg



Riff 42 Big.jpg



Riff 43 Big.jpg



Riff 44 Big.jpg



Riff 45 Big.jpg



Riff 46 Big.jpg



Riff 47 Big.jpg



Riff 48 Big.jpg



Riff 49 Big.jpg



Riff 50 Big.jpg



Riff 51 Big.jpg



Riff 52 Big.jpg



Riff 53 Big.jpg



Riff 54 Big.jpg



Riff 55 Big.jpg



Riff 56 Big.jpg



Riff 57 Big.jpg



Riff 58 Big.jpg



Riff 59 Big.jpg



Riff 60 Big.jpg



A062860

  • Smallest j with n nodes in its riff (rooted index-functional forest).

 



Riff 2 Big.jpg



Riff 3 Big.jpg



Riff 5 Big.jpg



Riff 10 Big.jpg



Riff 15 Big.jpg



Riff 30 Big.jpg



Riff 55 Big.jpg



Riff 105 Big.jpg



Riff 165 Big.jpg



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
Rote 802701 Big.jpg


Rote 1 Big.jpg



Rote 2 Big.jpg



Rote 3 Big.jpg



Rote 4 Big.jpg



Rote 5 Big.jpg



Rote 6 Big.jpg



Rote 7 Big.jpg



Rote 8 Big.jpg



Rote 9 Big.jpg



Rote 10 Big.jpg



Rote 11 Big.jpg



Rote 12 Big.jpg



Rote 13 Big.jpg



Rote 14 Big.jpg



Rote 15 Big.jpg



Rote 16 Big.jpg