User:Jon Awbrey/Riffs and Rotes

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Author: Jon Awbrey

Idea

Let \text{p}_{i}\! be the i^\text{th}\! prime, where the positive integer i\! is called the index of the prime \text{p}_{i}\! and the indices are taken in such a way that \text{p}_1 = 2.\! Thus the sequence of primes begins as follows:

\begin{matrix}
\text{p}_1 = 2,  &
\text{p}_2 = 3,  &
\text{p}_3 = 5,  &
\text{p}_4 = 7,  &
\text{p}_5 = 11, &
\text{p}_6 = 13, &
\text{p}_7 = 17, &
\text{p}_8 = 19, &
\ldots
\end{matrix}\!

The prime factorization of a positive integer n\! can be written in the following form:

n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},\!

where \text{p}_{i(k)}^{j(k)}\! is the k^\text{th}\! prime power in the factorization and \ell\! is the number of distinct prime factors dividing n.\! The factorization of 1\! is defined as 1\! in accord with the convention that an empty product is equal to 1.\!

Let I(n)\! be the set of indices of primes that divide n\! and let j(i, n)\! be the number of times that \text{p}_{i}\! divides n.\! Then the prime factorization of n\! can be written in the following alternative form:

n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.\!

For example:

\begin{matrix}
123456789
& = & 3^2 \cdot 3607 \cdot 3803
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1.
\end{matrix}\!

Each index i\! and exponent j\! appearing in the prime factorization of a positive integer n\! is itself a positive integer, and thus has a prime factorization of its own.

Continuing with the same example, the index 504\! has the factorization 2^3 \cdot 3^2 \cdot 7 = \text{p}_1^3 \text{p}_2^2 \text{p}_4^1\! and the index 529\! has the factorization {23}^2 = \text{p}_9^2.\! Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:

\begin{array}{rcl}
2 & \mapsto & \text{p}_1^1
\\[6pt]
504 & \mapsto & \text{p}_1^3 \text{p}_2^2 \text{p}_4^1
\\[6pt]
529 & \mapsto & \text{p}_9^2
\end{array}\!

This leads to the following development:

\begin{array}{lll}
123456789
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1
\end{array}\!

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

\begin{array}{lll}
123456789
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1
\\[18pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1
\\[18pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_2^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^2}^1}^1 \text{p}_{\text{p}_{\text{p}_2^2}^{\text{p}_1^1}}^1
\\[18pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^{\text{p}_1^1}}^1
\end{array}\!

The 1\!'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:

\begin{array}{lll}
123456789
& = & \text{p}_{\text{p}}^{\text{p}} \text{p}_{\text{p}^{\text{p}_{\text{p}}} \text{p}_{\text{p}}^{\text{p}} \text{p}_{\text{p}^{\text{p}}}} \text{p}_{\text{p}_{\text{p}_{\text{p}}^{\text{p}}}^{\text{p}}}
\end{array}\!

The pattern of indices and exponents illustrated here is called a doubly recursive factorization, or DRF. Applying the same procedure to any positive integer n\! produces an expression called the DRF of n.\!   If \mathbb{M}\! is the set of positive integers, \mathcal{L}\! is the set of DRF expressions, and the mapping defined by the factorization process is denoted \operatorname{drf} : \mathbb{M} \to \mathcal{L},\! then the doubly recursive factorization of n\! is denoted \operatorname{drf}(n).\!

The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called riffs and rotes, respectively.

\operatorname{riff}(123456789)\! is the following digraph:
Riff 123456789 Big.jpg
\operatorname{rote}(123456789)\! is the following graph:
Rote 123456789 Big.jpg

Riffs in Numerical Order

\text{Riffs in Numerical Order}\!

 


1\!


\begin{array}{l} \varnothing \\ 1 \end{array}\!

Riff 2 Big.jpg


\text{p}\!


\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\!

Riff 3 Big.jpg


\text{p}_\text{p}\!


\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\!

Riff 4 Big.jpg


\text{p}^\text{p}\!


\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\!

Riff 5 Big.jpg


\text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\!

Riff 6 Big ISW.jpg


\text{p} \text{p}_\text{p}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}~\!

Riff 7 Big.jpg


\text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\!

Riff 8 Big.jpg


\text{p}^{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\!

Riff 9 Big.jpg


\text{p}_\text{p}^\text{p}~\!


\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\!

Riff 10 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\!

Riff 11 Big.jpg


\text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\!

Riff 12 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}\!


\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\!

Riff 13 Big.jpg


\text{p}_{\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\!

Riff 14 Big.jpg


\text{p} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}~\!

Riff 15 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\!

Riff 16 Big.jpg


\text{p}^{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\!

Riff 17 Big ISW.jpg


\text{p}_{\text{p}_{\text{p}^\text{p}}}\!


\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\!

Riff 18 Big.jpg


\text{p} \text{p}_\text{p}^\text{p}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\!

Riff 19 Big.jpg


\text{p}_{\text{p}^{\text{p}_\text{p}}}\!


\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\!

Riff 20 Big ISW.jpg


\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\!

Riff 21 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}~\!


\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\!

Riff 22 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\!

Riff 23 Big.jpg


\text{p}_{\text{p}_\text{p}^\text{p}}\!


\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\!

Riff 24 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!


\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\!

Riff 25 Big.jpg


\text{p}_{\text{p}_\text{p}}^\text{p}\!


\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\!

Riff 26 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\!

Riff 27 Big.jpg


\text{p}_\text{p}^{\text{p}_\text{p}}\!


\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\!

Riff 28 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\!

Riff 29 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\!

Riff 30 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\!

Riff 31 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!


\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\!

Riff 32 Big.jpg


\text{p}^{\text{p}_{\text{p}_\text{p}}}~\!


\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\!

Riff 33 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\!

Riff 34 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\!

Riff 35 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\!

Riff 36 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!


\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\!

Riff 37 Big.jpg


\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\!

Riff 38 Big.jpg


\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}~\!

Riff 39 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\!

Riff 40 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\!

Riff 41 Big.jpg


\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!


\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\!

Riff 42 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\!

Riff 43 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!


\begin{array}{l} 14\!:\!1 \\ 43 \end{array}~\!

Riff 44 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\!

Riff 45 Big.jpg


\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}~\!

Riff 46 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\!

Riff 47 Big.jpg


\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\!

Riff 48 Big.jpg


\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!


\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\!

Riff 49 Big.jpg


\text{p}_{\text{p}^\text{p}}^\text{p}\!


\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\!

Riff 50 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!


\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\!

Riff 51 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\!

Riff 52 Big ISW.jpg


\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\!

Riff 53 Big.jpg


\text{p}_{\text{p}^{\text{p}^\text{p}}}\!


\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\!

Riff 54 Big.jpg


\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\!

Riff 55 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\!

Riff 56 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\!

Riff 57 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\!

Riff 58 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\!

Riff 59 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!


\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\!

Riff 60 Big.jpg


\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\!

Rotes in Numerical Order

\text{Rotes in Numerical Order}\!

Rote 1 Big.jpg


1\!


\begin{array}{l} \varnothing \\ 1 \end{array}\!

Rote 2 Big.jpg


\text{p}\!


\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\!

Rote 3 Big.jpg


\text{p}_\text{p}\!


\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\!

Rote 4 Big.jpg


\text{p}^\text{p}\!


\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\!

Rote 5 Big.jpg


\text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\!

Rote 6 Big.jpg


\text{p} \text{p}_\text{p}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}~\!

Rote 7 Big.jpg


\text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\!

Rote 8 Big.jpg


\text{p}^{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\!

Rote 9 Big.jpg


\text{p}_\text{p}^\text{p}~\!


\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\!

Rote 10 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\!

Rote 11 Big.jpg


\text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\!

Rote 12 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}\!


\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\!

Rote 13 Big.jpg


\text{p}_{\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\!

Rote 14 Big.jpg


\text{p} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}~\!

Rote 15 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\!

Rote 16 Big.jpg


\text{p}^{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\!

Rote 17 Big.jpg


\text{p}_{\text{p}_{\text{p}^\text{p}}}\!


\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\!

Rote 18 Big ISW.jpg


\text{p} \text{p}_\text{p}^\text{p}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\!

Rote 19 Big.jpg


\text{p}_{\text{p}^{\text{p}_\text{p}}}\!


\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\!

Rote 20 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\!

Rote 21 Big ISW.jpg


\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}~\!


\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\!

Rote 22 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\!

Rote 23 Big.jpg


\text{p}_{\text{p}_\text{p}^\text{p}}\!


\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\!

Rote 24 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!


\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\!

Rote 25 Big.jpg


\text{p}_{\text{p}_\text{p}}^\text{p}\!


\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\!

Rote 26 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\!

Rote 27 Big.jpg


\text{p}_\text{p}^{\text{p}_\text{p}}\!


\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\!

Rote 28 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\!

Rote 29 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\!

Rote 30 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\!

Rote 31 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!


\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\!

Rote 32 Big.jpg


\text{p}^{\text{p}_{\text{p}_\text{p}}}~\!


\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\!

Rote 33 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\!

Rote 34 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\!

Rote 35 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\!

Rote 36 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!


\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\!

Rote 37 Big.jpg


\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\!

Rote 38 Big.jpg


\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}~\!

Rote 39 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\!

Rote 40 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\!

Rote 41 Big.jpg


\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!


\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\!

Rote 42 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\!

Rote 43 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!


\begin{array}{l} 14\!:\!1 \\ 43 \end{array}~\!

Rote 44 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\!

Rote 45 Big.jpg


\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}~\!

Rote 46 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\!

Rote 47 Big.jpg


\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\!

Rote 48 Big.jpg


\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!


\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\!

Rote 49 Big.jpg


\text{p}_{\text{p}^\text{p}}^\text{p}\!


\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\!

Rote 50 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!


\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\!

Rote 51 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\!

Rote 52 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\!

Rote 53 Big.jpg


\text{p}_{\text{p}^{\text{p}^\text{p}}}\!


\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\!

Rote 54 Big.jpg


\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\!

Rote 55 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\!

Rote 56 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\!

Rote 57 Big ISW.jpg


\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\!

Rote 58 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\!

Rote 59 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!


\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\!

Rote 60 Big.jpg


\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\!

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.
\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!
\text{Integer}\! \text{Factorization}\! \text{Notation}\! \text{Riff Digraph}\! \text{Rote Graph}\! \text{Traversal}\!
1\! 1\!     Rote 1 Big.jpg  
2\! \text{p}_1^1\! \text{p}\! Riff 2 Big.jpg Rote 2 Big.jpg ((~))\!
3\!

\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}\!

\text{p}_\text{p}\! Riff 3 Big.jpg Rote 3 Big.jpg (((~))(~))\!
4\!

\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}\!

\text{p}^\text{p}\! Riff 4 Big.jpg Rote 4 Big.jpg ((((~))))\!
5\!

\begin{array}{lll}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\\[10pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}\!

\text{p}_{\text{p}_{\text{p}}}\! Riff 5 Big.jpg Rote 5 Big.jpg ((((~))(~))(~))\!
6\!

\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}\!

\text{p} \text{p}_{\text{p}}\! Riff 6 Big ISW.jpg Rote 6 Big.jpg ((~))(((~))(~))\!
7\!

\begin{array}{lll}
\text{p}_4^1
& = & \text{p}_{\text{p}_1^2}^1
\\[10pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}\!

\text{p}_{\text{p}^{\text{p}}}\! Riff 7 Big.jpg Rote 7 Big.jpg (((((~))))(~))\!
8\!

\begin{array}{lll}
\text{p}_1^3
& = & \text{p}_1^{\text{p}_2^1}
\\[10pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}\!

\text{p}^{\text{p}_{\text{p}}}\,\! Riff 8 Big.jpg Rote 8 Big.jpg (((((~))(~))))\!
9\!

\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}\!

\text{p}_\text{p}^\text{p}\,\! Riff 9 Big.jpg Rote 9 Big.jpg (((~))(((~))))\!
16\!

\begin{array}{lll}
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
\\[10pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}\!

\text{p}^{\text{p}^{\text{p}}}\! 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.

\begin{array}{l|l|r}
k
& P_k
= \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \}
= \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \}
& |P_k|
\\[10pt]
0 & \{ 1 \} & 1
\\
1 & \{ 2 \} & 1
\\
2 & \{ 3, 4 \} & 2
\\
3 & \{ 5, 6, 7, 8, 9, 16 \} & 6
\\
4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20
\end{array}\!

\text{Prime Factorizations, Riffs, and Rotes}\!
\text{Integer}\! \text{Factorization}\! \text{Notation}\! \text{Riff Digraph}\! \text{Rote Graph}\!
1\! 1\!     Rote 1 Big.jpg
2\! \text{p}_1^1\! \text{p}\! Riff 2 Big.jpg Rote 2 Big.jpg
3\!

\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}\!

\text{p}_\text{p}\! Riff 3 Big.jpg Rote 3 Big.jpg
4\!

\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}\!

\text{p}^\text{p}\! Riff 4 Big.jpg Rote 4 Big.jpg
5\!

\begin{array}{lll}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}\!

\text{p}_{\text{p}_{\text{p}}}\! Riff 5 Big.jpg Rote 5 Big.jpg
6\!

\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}\!

\text{p} \text{p}_{\text{p}}\! Riff 6 Big ISW.jpg Rote 6 Big.jpg
7\!

\begin{array}{lll}
\text{p}_4^1
& = & \text{p}_{\text{p}_1^2}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}\!

\text{p}_{\text{p}^{\text{p}}}\! Riff 7 Big.jpg Rote 7 Big.jpg
8\!

\begin{array}{lll}
\text{p}_1^3
& = & \text{p}_1^{\text{p}_2^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}\!

\text{p}^{\text{p}_{\text{p}}}\,\! Riff 8 Big.jpg Rote 8 Big.jpg
9\!

\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}\!

\text{p}_\text{p}^\text{p}\,\! Riff 9 Big.jpg Rote 9 Big.jpg
16\!

\begin{array}{lll}
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}\!

\text{p}^{\text{p}^{\text{p}}}\! Riff 16 Big.jpg Rote 16 Big.jpg
10\!

\begin{array}{lll}
\text{p}_1^1 \text{p}_3^1
& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1
\\[12pt]
& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}\!

\text{p} \text{p}_{\text{p}_{\text{p}}}\! Riff 10 Big.jpg Rote 10 Big.jpg
11\!

\begin{array}{lll}
\text{p}_5^1
& = & \text{p}_{\text{p}_3^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1
\end{array}\!

\text{p}_{\text{p}_{\text{p}_\text{p}}}\! Riff 11 Big.jpg Rote 11 Big.jpg
12\!

\begin{array}{lll}
\text{p}_1^2 \text{p}_2^1
& = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1
\end{array}\!

\text{p}^{\text{p}} \text{p}_{\text{p}}\! Riff 12 Big.jpg Rote 12 Big.jpg
13\!

\begin{array}{lll}
\text{p}_6^1
& = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1
\end{array}\!

\text{p}_{\text{p} \text{p}_{\text{p}}}\! Riff 13 Big.jpg Rote 13 Big.jpg
14\!

\begin{array}{lll}
\text{p}_1^1 \text{p}_4^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1
\\[12pt]
& = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}\!

\text{p} \text{p}_{\text{p}^{\text{p}}}\! Riff 14 Big.jpg Rote 14 Big.jpg
17\!

\begin{array}{lll}
\text{p}_7^1
& = & \text{p}_{\text{p}_4^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
\end{array}\!

\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\! Riff 17 Big ISW.jpg Rote 17 Big.jpg
18\!

\begin{array}{lll}
\text{p}_1^1 \text{p}_2^2
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}\!

\text{p} \text{p}_{\text{p}}^{\text{p}}\! Riff 18 Big.jpg Rote 18 Big ISW.jpg
19~\!

\begin{array}{lll}
\text{p}_8^1
& = & \text{p}_{\text{p}_1^3}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1
\end{array}\!

\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\! Riff 19 Big.jpg Rote 19 Big.jpg
23\!

\begin{array}{lll}
\text{p}_9^1
& = & \text{p}_{\text{p}_2^2}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1
\end{array}\!

\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\! Riff 23 Big.jpg Rote 23 Big.jpg
25\!

\begin{array}{lll}
\text{p}_3^2
& = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1}
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1}
\end{array}\!

\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\! Riff 25 Big.jpg Rote 25 Big.jpg
27\!

\begin{array}{lll}
\text{p}_2^3
& = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1}
\\[12pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1}
\end{array}\!

\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\! Riff 27 Big.jpg Rote 27 Big.jpg
32\!

\begin{array}{lll}
\text{p}_1^5
& = & \text{p}_1^{\text{p}_3^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}
\end{array}\!

\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\! Riff 32 Big.jpg Rote 32 Big.jpg
49\!

\begin{array}{lll}
\text{p}_4^2
& = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1}
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1}
\end{array}\!

\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\! Riff 49 Big.jpg Rote 49 Big.jpg
53\!

\begin{array}{lll}
\text{p}_{16}^1
& = & \text{p}_{\text{p}_1^4}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1
\end{array}\!

\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\! Riff 53 Big.jpg Rote 53 Big.jpg
64\!

\begin{array}{lll}
\text{p}_1^6
& = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
\end{array}\!

\text{p}^{\text{p} \text{p}_{\text{p}}}\! Riff 64 Big.jpg Rote 64 Big.jpg
81\!

\begin{array}{lll}
\text{p}_2^4
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2}
\\[12pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}}
\end{array}\!

\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\! Riff 81 Big.jpg Rote 81 Big.jpg
128\!

\begin{array}{lll}
\text{p}_1^7
& = & \text{p}_1^{\text{p}_4^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}
\end{array}\!

\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\! Riff 128 Big.jpg Rote 128 Big.jpg
256\!

\begin{array}{lll}
\text{p}_1^8
& = & \text{p}_1^{\text{p}_1^3}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}
\end{array}\!

\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\! Riff 256 Big.jpg Rote 256 Big.jpg
512\!

\begin{array}{lll}
\text{p}_1^9
& = & \text{p}_1^{\text{p}_2^2}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}
\end{array}\!

\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\! Riff 512 Big.jpg Rote 512 Big.jpg
65536\!

\begin{array}{lll}
\text{p}_1^{16}
& = & \text{p}_1^{\text{p}_1^4}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}
\end{array}\!

\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\! Riff 65536 Big.jpg Rote 65536 Big.jpg

A062537

  • Nodes in riff (rooted index-functional forest) for n.
a(n) = \text{Number of Nodes in the Riff of}~ n\!

 


1\!


a(1) ~=~ 0\!

Riff 2 Big.jpg


\text{p}\!


a(2) ~=~ 1\!

Riff 3 Big.jpg


\text{p}_\text{p}\!


a(3) ~=~ 2\!

Riff 4 Big.jpg


\text{p}^\text{p}\!


a(4) ~=~ 2\!

Riff 5 Big.jpg


\text{p}_{\text{p}_{\text{p}}}\!


a(5) ~=~ 3\!

Riff 6 Big ISW.jpg


\text{p} \text{p}_{\text{p}}\!


a(6) ~=~ 3\!

Riff 7 Big.jpg


\text{p}_{\text{p}^{\text{p}}}\!


a(7) ~=~ 3\!

Riff 8 Big.jpg


\text{p}^{\text{p}_{\text{p}}}\,\!


a(8) ~=~ 3\!

Riff 9 Big.jpg


\text{p}_\text{p}^\text{p}\,\!


a(9) ~=~ 3\!

Riff 10 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}}}\!


a(10) ~=~ 4\!

Riff 11 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!


a(11) ~=~ 4\!

Riff 12 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}\!


a(12) ~=~ 4\!

Riff 13 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}}}\!


a(13) ~=~ 4\!

Riff 14 Big.jpg


\text{p} \text{p}_{\text{p}^{\text{p}}}\!


a(14) ~=~ 4\!

Riff 15 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!


a(15) ~=~ 5\!

Riff 16 Big.jpg


\text{p}^{\text{p}^{\text{p}}}\!


a(16) ~=~ 3\!

Riff 17 Big ISW.jpg


\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!


a(17) ~=~ 4\!

Riff 18 Big.jpg


\text{p} \text{p}_\text{p}^\text{p}\!


a(18) ~=~ 4~\!

Riff 19 Big.jpg


\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!


a(19) ~=~ 4\!

Riff 20 Big ISW.jpg


\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!


a(20) ~=~ 5\!

Riff 21 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}~\!


a(21) ~=~ 5\!

Riff 22 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(22) ~=~ 5\!

Riff 23 Big.jpg


\text{p}_{\text{p}_\text{p}^\text{p}}\!


a(23) ~=~ 4\!

Riff 24 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!


a(24) ~=~ 5\!

Riff 25 Big.jpg


\text{p}_{\text{p}_\text{p}}^\text{p}\!


a(25) ~=~ 4\,\!

Riff 26 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(26) ~=~ 5\!

Riff 27 Big.jpg


\text{p}_\text{p}^{\text{p}_\text{p}}\!


a(27) ~=~ 4\!

Riff 28 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!


a(28) ~=~ 5\!

Riff 29 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(29) ~=~ 5\!

Riff 30 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(30) ~=~ 6\!

Riff 31 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!


a(31) ~=~ 5\!

Riff 32 Big.jpg


\text{p}^{\text{p}_{\text{p}_\text{p}}}~\!


a(32) ~=~ 4\!

Riff 33 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(33) ~=~ 6\!

Riff 34 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


a(34) ~=~ 5\!

Riff 35 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


a(35) ~=~ 6\!

Riff 36 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!


a(36) ~=~ 5\!

Riff 37 Big.jpg


\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!


a(37) ~=~ 5\!

Riff 38 Big.jpg


\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


a(38) ~=~ 5\!

Riff 39 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(39) ~=~ 6\!

Riff 40 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!


a(40) ~=~ 6\!

Riff 41 Big.jpg


\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!


a(41) ~=~ 5\!

Riff 42 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!


a(42) ~=~ 6\!

Riff 43 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!


a(43) ~=~ 5\!

Riff 44 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(44) ~=~ 6\!

Riff 45 Big.jpg


\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


a(45) ~=~ 6\!

Riff 46 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!


a(46) ~=~ 5\!

Riff 47 Big.jpg


\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(47) ~=~ 6\!

Riff 48 Big.jpg


\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!


a(48) ~=~ 5\!

Riff 49 Big.jpg


\text{p}_{\text{p}^\text{p}}^\text{p}\!


a(49) ~=~ 4\!

Riff 50 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!


a(50) ~=~ 5\!

Riff 51 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


a(51) ~=~ 6\!

Riff 52 Big ISW.jpg


\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(52) ~=~ 6\!

Riff 53 Big.jpg


\text{p}_{\text{p}^{\text{p}^\text{p}}}\!


a(53) ~=~ 4\!

Riff 54 Big.jpg


\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!


a(54) ~=~ 5\!

Riff 55 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(55) ~=~ 7\!

Riff 56 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


a(56) ~=~ 6\!

Riff 57 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


a(57) ~=~ 6\!

Riff 58 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(58) ~=~ 6\!

Riff 59 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!


a(59) ~=~ 5\!

Riff 60 Big.jpg


\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(60) ~=~ 7\!

A062860

  • Smallest j with n nodes in its riff (rooted index-functional forest).
a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}\!

 


1\!


a(0) ~=~ 1\!

Riff 2 Big.jpg


\text{p}\!


a(1) ~=~ 2\!

Riff 3 Big.jpg


\text{p}_\text{p}\!


a(2) ~=~ 3\!

Riff 5 Big.jpg


\text{p}_{\text{p}_{\text{p}}}\!


a(3) ~=~ 5\!

Riff 10 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}}}\!


a(4) ~=~ 10\!

Riff 15 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!


a(5) ~=~ 15\!

Riff 30 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(6) ~=~ 30\!

Riff 55 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(7) ~=~ 55\!

Riff 105 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


a(8) ~=~ 105\!

Riff 165 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(9) ~=~ 165\!

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
802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1\!
\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}\!
\begin{array}{lllll}
802701
& = & 9 \cdot 89189
& = & 2\!:\!2 ~~ 8638\!:\!1
\\
8638
& = & 2 \cdot 7 \cdot 617
& = & 1\!:\!1 ~~ 4\!:\!1 ~~ 113\!:\!1
\\
113
&   &
& = & 30\!:\!1
\\
30
& = & 2 \cdot 3 \cdot 5
& = & 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1
\\
4
&   &
& = & 1\!:\!2
\\
3
&   &
& = & 2\!:\!1
\\
2
&   &
& = & 1\!:\!1
\end{array}\!
\text{So the rote of 802701 is the following graph:}\!
Rote 802701 Big.jpg
\text{By inspection, the rote height of 802701 is 6.}\!


a(n) = \text{Rote Height of}~ n~\!

Rote 1 Big.jpg


1\!


a(1) ~=~ 0\!

Rote 2 Big.jpg


\text{p}\!


a(2) ~=~ 1\!

Rote 3 Big.jpg


\text{p}_\text{p}\!


a(3) ~=~ 2\!

Rote 4 Big.jpg


\text{p}^\text{p}\!


a(4) ~=~ 2\!

Rote 5 Big.jpg


\text{p}_{\text{p}_\text{p}}\!


a(5) ~=~ 3\!

Rote 6 Big.jpg


\text{p} \text{p}_\text{p}\!


a(6) ~=~ 2\!

Rote 7 Big.jpg


\text{p}_{\text{p}^\text{p}}\!


a(7) ~=~ 3\!

Rote 8 Big.jpg


\text{p}^{\text{p}_\text{p}}\!


a(8) ~=~ 3\!

Rote 9 Big.jpg


\text{p}_\text{p}^\text{p}~\!


a(9) ~=~ 2\!

Rote 10 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}\!


a(10) ~=~ 3\!

Rote 11 Big.jpg


\text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(11) ~=~ 4\!

Rote 12 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}\!


a(12) ~=~ 2\!

Rote 13 Big.jpg


\text{p}_{\text{p} \text{p}_\text{p}}\!


a(13) ~=~ 3\!

Rote 14 Big.jpg


\text{p} \text{p}_{\text{p}^\text{p}}\!


a(14) ~=~ 3\!

Rote 15 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(15) ~=~ 3\!

Rote 16 Big.jpg


\text{p}^{\text{p}^\text{p}}\!


a(16) ~=~ 3\!

Rote 17 Big.jpg


\text{p}_{\text{p}_{\text{p}^\text{p}}}\!


a(17) ~=~ 4\!

Rote 18 Big ISW.jpg


\text{p} \text{p}_\text{p}^\text{p}\!


a(18) ~=~ 2\!

Rote 19 Big.jpg


\text{p}_{\text{p}^{\text{p}_\text{p}}}\!


a(19) ~=~ 4\!

Rote 20 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


a(20) ~=~ 3\!

Rote 21 Big ISW.jpg


\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}~\!


a(21) ~=~ 3\!

Rote 22 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(22) ~=~ 4\!

Rote 23 Big.jpg


\text{p}_{\text{p}_\text{p}^\text{p}}\!


a(23) ~=~ 3\!

Rote 24 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!


a(24) ~=~ 3\!

Rote 25 Big.jpg


\text{p}_{\text{p}_\text{p}}^\text{p}\!


a(25) ~=~ 3\!

Rote 26 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(26) ~=~ 3\!

Rote 27 Big.jpg


\text{p}_\text{p}^{\text{p}_\text{p}}\!


a(27) ~=~ 3\!

Rote 28 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!


a(28) ~=~ 3\!

Rote 29 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(29) ~=~ 4\!

Rote 30 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(30) ~=~ 3\!

Rote 31 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!


a(31) ~=~ 5\!

Rote 32 Big.jpg


\text{p}^{\text{p}_{\text{p}_\text{p}}}~\!


a(32) ~=~ 4\!

Rote 33 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(33) ~=~ 4\!

Rote 34 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


a(34) ~=~ 4\!

Rote 35 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


a(35) ~=~ 3\!

Rote 36 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!


a(36) ~=~ 2\!

Rote 37 Big.jpg


\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!


a(37) ~=~ 3\!

Rote 38 Big.jpg


\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


a(38) ~=~ 4\!

Rote 39 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(39) ~=~ 3\!

Rote 40 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!


a(40) ~=~ 3\!

Rote 41 Big.jpg


\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!


a(41) ~=~ 4\!

Rote 42 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!


a(42) ~=~ 3\!

Rote 43 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!


a(43) ~=~ 4\!

Rote 44 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(44) ~=~ 4\!

Rote 45 Big.jpg


\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


a(45) ~=~ 3\!

Rote 46 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!


a(46) ~=~ 3\!

Rote 47 Big.jpg


\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(47) ~=~ 4\!

Rote 48 Big.jpg


\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!


a(48) ~=~ 3\!

Rote 49 Big.jpg


\text{p}_{\text{p}^\text{p}}^\text{p}\!


a(49) ~=~ 3\!

Rote 50 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!


a(50) ~=~ 3\!

Rote 51 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


a(51) ~=~ 4\!

Rote 52 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(52) ~=~ 3\!

Rote 53 Big.jpg


\text{p}_{\text{p}^{\text{p}^\text{p}}}\!


a(53) ~=~ 4\!

Rote 54 Big.jpg


\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!


a(54) ~=~ 3\!

Rote 55 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(55) ~=~ 4\!

Rote 56 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


a(56) ~=~ 3\!

Rote 57 Big ISW.jpg


\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


a(57) ~=~ 4\!

Rote 58 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(58) ~=~ 4\!

Rote 59 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!


a(59) ~=~ 5\!

Rote 60 Big.jpg


\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(60) ~=~ 3\!

Miscellaneous Examples

\text{Integers, Riffs, Rotes}\!
\text{Integer}\! \text{Riff}\! \text{Rote}\!
1\!   Rote 1 Big.jpg
2\! Riff 2 Big.jpg Rote 2 Big.jpg
3\! Riff 3 Big.jpg Rote 3 Big.jpg
4\! Riff 4 Big.jpg Rote 4 Big.jpg
360\! Riff 360 Big.jpg Rote 360 Big.jpg
2010\! Riff 2010 Big.jpg Rote 2010 Big.jpg
2011\! Riff 2011 Big.jpg Rote 2011 Big.jpg
2012\! Riff 2012 Big.jpg Rote 2012 Big.jpg
2017\! Riff 2017 Big.jpg Rote 2017 Big.jpg
2500\! Riff 2500 Big.jpg Rote 2500 Big.jpg
802701\! Riff 802701 Big.jpg Rote 802701 Big.jpg
123456789\! Riff 123456789 Big.jpg Rote 123456789 Big.jpg