This site is supported by donations to The OEIS Foundation.

# Riffs and Rotes

## 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: $\operatorname{rote}(123456789)$ is the following graph:

## Riffs in Numerical Order

 $1\!$ $\begin{array}{l} \varnothing \\ 1 \end{array}$ $\text{p}\!$ $\begin{array}{l} 1\!:\!1 \\ 2 \end{array}$ $\text{p}_\text{p}\!$ $\begin{array}{l} 2\!:\!1 \\ 3 \end{array}$ $\text{p}^\text{p}\!$ $\begin{array}{l} 1\!:\!2 \\ 4 \end{array}$ $\text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 3\!:\!1 \\ 5 \end{array}$ $\text{p} \text{p}_\text{p}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}$ $\text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 4\!:\!1 \\ 7 \end{array}$ $\text{p}^{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!3 \\ 8 \end{array}$ $\text{p}_\text{p}^\text{p}\!$ $\begin{array}{l} 2\!:\!2 \\ 9 \end{array}$ $\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}$ $\text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 5\!:\!1 \\ 11 \end{array}$ $\text{p}^\text{p} \text{p}_\text{p}\!$ $\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}$ $\text{p}_{\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 6\!:\!1 \\ 13 \end{array}$ $\text{p} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}$ $\text{p}^{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!4 \\ 16 \end{array}$ $\text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $\begin{array}{l} 7\!:\!1 \\ 17 \end{array}$ $\text{p} \text{p}_\text{p}^\text{p}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}$ $\text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $\begin{array}{l} 8\!:\!1 \\ 19 \end{array}$ $\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}$ $\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}$ $\text{p}_{\text{p}_\text{p}^\text{p}}\!$ $\begin{array}{l} 9\!:\!1 \\ 23 \end{array}$ $\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!$ $\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}$ $\text{p}_{\text{p}_\text{p}}^\text{p}\!$ $\begin{array}{l} 3\!:\!2 \\ 25 \end{array}$ $\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}$ $\text{p}_\text{p}^{\text{p}_\text{p}}\!$ $\begin{array}{l} 2\!:\!3 \\ 27 \end{array}$ $\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}$ $\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 10\!:\!1 \\ 29 \end{array}$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}$ $\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!$ $\begin{array}{l} 11\!:\!1 \\ 31 \end{array}$ $\text{p}^{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!5 \\ 32 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}$ $\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}$ $\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!$ $\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}$ $\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 12\!:\!1 \\ 37 \end{array}$ $\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}$ $\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!$ $\begin{array}{l} 13\!:\!1 \\ 41 \end{array}$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}$ $\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!$ $\begin{array}{l} 14\!:\!1 \\ 43 \end{array}$ $\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}$ $\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}$ $\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}$ $\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 15\!:\!1 \\ 47 \end{array}$ $\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!$ $\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}$ $\text{p}_{\text{p}^\text{p}}^\text{p}\!$ $\begin{array}{l} 4\!:\!2 \\ 49 \end{array}$ $\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!$ $\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}$ $\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}$ $\text{p}_{\text{p}^{\text{p}^\text{p}}}\!$ $\begin{array}{l} 16\!:\!1 \\ 53 \end{array}$ $\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}$ $\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}$ $\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!$ $\begin{array}{l} 17\!:\!1 \\ 59 \end{array}$ $\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}$

## Rotes in Numerical Order

 $1\!$ $\begin{array}{l} \varnothing \\ 1 \end{array}$ $\text{p}\!$ $\begin{array}{l} 1\!:\!1 \\ 2 \end{array}$ $\text{p}_\text{p}\!$ $\begin{array}{l} 2\!:\!1 \\ 3 \end{array}$ $\text{p}^\text{p}\!$ $\begin{array}{l} 1\!:\!2 \\ 4 \end{array}$ $\text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 3\!:\!1 \\ 5 \end{array}$ $\text{p} \text{p}_\text{p}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}$ $\text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 4\!:\!1 \\ 7 \end{array}$ $\text{p}^{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!3 \\ 8 \end{array}$ $\text{p}_\text{p}^\text{p}\!$ $\begin{array}{l} 2\!:\!2 \\ 9 \end{array}$ $\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}$ $\text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 5\!:\!1 \\ 11 \end{array}$ $\text{p}^\text{p} \text{p}_\text{p}\!$ $\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}$ $\text{p}_{\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 6\!:\!1 \\ 13 \end{array}$ $\text{p} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}$ $\text{p}^{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!4 \\ 16 \end{array}$ $\text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $\begin{array}{l} 7\!:\!1 \\ 17 \end{array}$ $\text{p} \text{p}_\text{p}^\text{p}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}$ $\text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $\begin{array}{l} 8\!:\!1 \\ 19 \end{array}$ $\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}$ $\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}$ $\text{p}_{\text{p}_\text{p}^\text{p}}\!$ $\begin{array}{l} 9\!:\!1 \\ 23 \end{array}$ $\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!$ $\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}$ $\text{p}_{\text{p}_\text{p}}^\text{p}\!$ $\begin{array}{l} 3\!:\!2 \\ 25 \end{array}$ $\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}$ $\text{p}_\text{p}^{\text{p}_\text{p}}\!$ $\begin{array}{l} 2\!:\!3 \\ 27 \end{array}$ $\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}$ $\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 10\!:\!1 \\ 29 \end{array}$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}$ $\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!$ $\begin{array}{l} 11\!:\!1 \\ 31 \end{array}$ $\text{p}^{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!5 \\ 32 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}$ $\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}$ $\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!$ $\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}$ $\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 12\!:\!1 \\ 37 \end{array}$ $\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}$ $\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!$ $\begin{array}{l} 13\!:\!1 \\ 41 \end{array}$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}$ $\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!$ $\begin{array}{l} 14\!:\!1 \\ 43 \end{array}$ $\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}$ $\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}$ $\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}$ $\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 15\!:\!1 \\ 47 \end{array}$ $\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!$ $\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}$ $\text{p}_{\text{p}^\text{p}}^\text{p}\!$ $\begin{array}{l} 4\!:\!2 \\ 49 \end{array}$ $\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!$ $\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}$ $\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}$ $\text{p}_{\text{p}^{\text{p}^\text{p}}}\!$ $\begin{array}{l} 16\!:\!1 \\ 53 \end{array}$ $\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}$ $\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}$ $\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}$ $\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!$ $\begin{array}{l} 17\!:\!1 \\ 59 \end{array}$ $\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}$

## 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.
$\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!$
 $\text{Integer}\!$ $\text{Factorization}\!$ $\text{Notation}\!$ $\text{Riff Digraph}\!$ $\text{Rote Graph}\!$ $\text{Traversal}\!$
 $1\!$ $1\!$
 $2\!$ $\text{p}_1^1\!$ $\text{p}\!$ $((~))$
 $3\!$ $\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}$ $\text{p}_\text{p}\!$ $(((~))(~))$ $4\!$ $\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}$ $\text{p}^\text{p}\!$ $((((~))))$
 $5\!$ $\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[10pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}$ $\text{p}_{\text{p}_{\text{p}}}\!$ $((((~))(~))(~))$ $6\!$ $\begin{array}{lll} \text{p}_1^1 \text{p}_2^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 \end{array}$ $\text{p} \text{p}_{\text{p}}\!$ $((~))(((~))(~))$ $7\!$ $\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[10pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}$ $\text{p}_{\text{p}^{\text{p}}}\!$ $(((((~))))(~))$ $8\!$ $\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[10pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}$ $\text{p}^{\text{p}_{\text{p}}}\!$ $(((((~))(~))))$ $9\!$ $\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}$ $\text{p}_\text{p}^\text{p}\!$ $(((~))(((~))))$ $16\!$ $\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[10pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}$ $\text{p}^{\text{p}^{\text{p}}}\!$ $((((((~))))))$

### A062504

• Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.
 $\begin{array}{l|l|r} k & P_k = \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \} = \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \} & |P_k| \\[10pt] 0 & \{ 1 \} & 1 \\ 1 & \{ 2 \} & 1 \\ 2 & \{ 3, 4 \} & 2 \\ 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 \\ 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 \end{array}$
$\text{Prime Factorizations, Riffs, and Rotes}\!$
 $\text{Integer}\!$ $\text{Factorization}\!$ $\text{Notation}\!$ $\text{Riff Digraph}\!$ $\text{Rote Graph}\!$
 $1\!$ $1\!$
 $2\!$ $\text{p}_1^1\!$ $\text{p}\!$
 $3\!$ $\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}$ $\text{p}_\text{p}\!$ $4\!$ $\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}$ $\text{p}^\text{p}\!$
 $5\!$ $\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}$ $\text{p}_{\text{p}_{\text{p}}}\!$ $6\!$ $\begin{array}{lll} \text{p}_1^1 \text{p}_2^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 \end{array}$ $\text{p} \text{p}_{\text{p}}\!$ $7\!$ $\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}$ $\text{p}_{\text{p}^{\text{p}}}\!$ $8\!$ $\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}$ $\text{p}^{\text{p}_{\text{p}}}\!$ $9\!$ $\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}$ $\text{p}_\text{p}^\text{p}\!$ $16\!$ $\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}$ $\text{p}^{\text{p}^{\text{p}}}\!$
 $10\!$ $\begin{array}{lll} \text{p}_1^1 \text{p}_3^1 & = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}$ $\text{p} \text{p}_{\text{p}_{\text{p}}}\!$ $11\!$ $\begin{array}{lll} \text{p}_5^1 & = & \text{p}_{\text{p}_3^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 \end{array}$ $\text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $12\!$ $\begin{array}{lll} \text{p}_1^2 \text{p}_2^1 & = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1 \end{array}$ $\text{p}^{\text{p}} \text{p}_{\text{p}}\!$ $13\!$ $\begin{array}{lll} \text{p}_6^1 & = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1 \\[12pt] & = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1 \end{array}$ $\text{p}_{\text{p} \text{p}_{\text{p}}}\!$ $14\!$ $\begin{array}{lll} \text{p}_1^1 \text{p}_4^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}$ $\text{p} \text{p}_{\text{p}^{\text{p}}}\!$ $17\!$ $\begin{array}{lll} \text{p}_7^1 & = & \text{p}_{\text{p}_4^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \end{array}$ $\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!$ $18\!$ $\begin{array}{lll} \text{p}_1^1 \text{p}_2^2 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}$ $\text{p} \text{p}_{\text{p}}^{\text{p}}\!$ $19\!$ $\begin{array}{lll} \text{p}_8^1 & = & \text{p}_{\text{p}_1^3}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1 \end{array}$ $\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!$ $23\!$ $\begin{array}{lll} \text{p}_9^1 & = & \text{p}_{\text{p}_2^2}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1 \end{array}$ $\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!$ $25\!$ $\begin{array}{lll} \text{p}_3^2 & = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1} \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1} \end{array}$ $\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!$ $27\!$ $\begin{array}{lll} \text{p}_2^3 & = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1} \\[12pt] & = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1} \end{array}$ $\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!$ $32\!$ $\begin{array}{lll} \text{p}_1^5 & = & \text{p}_1^{\text{p}_3^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1} \end{array}$ $\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!$ $49\!$ $\begin{array}{lll} \text{p}_4^2 & = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1} \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1} \end{array}$ $\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!$ $53\!$ $\begin{array}{lll} \text{p}_{16}^1 & = & \text{p}_{\text{p}_1^4}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1 \end{array}$ $\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!$ $64\!$ $\begin{array}{lll} \text{p}_1^6 & = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1} \\[12pt] & = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1} \end{array}$ $\text{p}^{\text{p} \text{p}_{\text{p}}}\!$ $81\!$ $\begin{array}{lll} \text{p}_2^4 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2} \\[12pt] & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}} \end{array}$ $\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!$ $128\!$ $\begin{array}{lll} \text{p}_1^7 & = & \text{p}_1^{\text{p}_4^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1} \end{array}$ $\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!$ $256\!$ $\begin{array}{lll} \text{p}_1^8 & = & \text{p}_1^{\text{p}_1^3} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}} \end{array}$ $\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!$ $512\!$ $\begin{array}{lll} \text{p}_1^9 & = & \text{p}_1^{\text{p}_2^2} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}} \end{array}$ $\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!$ $65536\!$ $\begin{array}{lll} \text{p}_1^{16} & = & \text{p}_1^{\text{p}_1^4} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}} \end{array}$ $\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!$

### A062537

• Nodes in riff (rooted index-functional forest) for n.
 $1\!$ $a(1) ~=~ 0$ $\text{p}\!$ $a(2) ~=~ 1$ $\text{p}_\text{p}\!$ $a(3) ~=~ 2$ $\text{p}^\text{p}\!$ $a(4) ~=~ 2$ $\text{p}_{\text{p}_{\text{p}}}\!$ $a(5) ~=~ 3$ $\text{p} \text{p}_{\text{p}}\!$ $a(6) ~=~ 3$ $\text{p}_{\text{p}^{\text{p}}}\!$ $a(7) ~=~ 3$ $\text{p}^{\text{p}_{\text{p}}}\!$ $a(8) ~=~ 3$ $\text{p}_\text{p}^\text{p}\!$ $a(9) ~=~ 3$ $\text{p} \text{p}_{\text{p}_{\text{p}}}\!$ $a(10) ~=~ 4$ $\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!$ $a(11) ~=~ 4$ $\text{p}^\text{p} \text{p}_\text{p}\!$ $a(12) ~=~ 4$ $\text{p}_{\text{p} \text{p}_{\text{p}}}\!$ $a(13) ~=~ 4$ $\text{p} \text{p}_{\text{p}^{\text{p}}}\!$ $a(14) ~=~ 4$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!$ $a(15) ~=~ 5$ $\text{p}^{\text{p}^{\text{p}}}\!$ $a(16) ~=~ 3$ $\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!$ $a(17) ~=~ 4$ $\text{p} \text{p}_\text{p}^\text{p}\!$ $a(18) ~=~ 4$ $\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!$ $a(19) ~=~ 4$ $\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!$ $a(20) ~=~ 5$ $\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(21) ~=~ 5$ $\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(22) ~=~ 5$ $\text{p}_{\text{p}_\text{p}^\text{p}}\!$ $a(23) ~=~ 4$ $\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!$ $a(24) ~=~ 5$ $\text{p}_{\text{p}_\text{p}}^\text{p}\!$ $a(25) ~=~ 4$ $\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(26) ~=~ 5$ $\text{p}_\text{p}^{\text{p}_\text{p}}\!$ $a(27) ~=~ 4$ $\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(28) ~=~ 5$ $\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(29) ~=~ 5$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(30) ~=~ 6$ $\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!$ $a(31) ~=~ 5$ $\text{p}^{\text{p}_{\text{p}_\text{p}}}\!$ $a(32) ~=~ 4$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(33) ~=~ 6$ $\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $a(34) ~=~ 5$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $a(35) ~=~ 6$ $\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!$ $a(36) ~=~ 5$ $\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!$ $a(37) ~=~ 5$ $\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $a(38) ~=~ 5$ $\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(39) ~=~ 6$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!$ $a(40) ~=~ 6$ $\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!$ $a(41) ~=~ 5$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(42) ~=~ 6$ $\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!$ $a(43) ~=~ 5$ $\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(44) ~=~ 6$ $\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(45) ~=~ 6$ $\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!$ $a(46) ~=~ 5$ $\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(47) ~=~ 6$ $\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!$ $a(48) ~=~ 5$ $\text{p}_{\text{p}^\text{p}}^\text{p}\!$ $a(49) ~=~ 4$ $\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!$ $a(50) ~=~ 5$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $a(51) ~=~ 6$ $\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(52) ~=~ 6$ $\text{p}_{\text{p}^{\text{p}^\text{p}}}\!$ $a(53) ~=~ 4$ $\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!$ $a(54) ~=~ 5$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(55) ~=~ 7$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $a(56) ~=~ 6$ $\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $a(57) ~=~ 6$ $\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(58) ~=~ 6$ $\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!$ $a(59) ~=~ 5$ $\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(60) ~=~ 7$

### A062860

• Smallest j with n nodes in its riff (rooted index-functional forest).
 $1\!$ $a(0) ~=~ 1$ $\text{p}\!$ $a(1) ~=~ 2$ $\text{p}_\text{p}\!$ $a(2) ~=~ 3$ $\text{p}_{\text{p}_{\text{p}}}\!$ $a(3) ~=~ 5$ $\text{p} \text{p}_{\text{p}_{\text{p}}}\!$ $a(4) ~=~ 10$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!$ $a(5) ~=~ 15$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(6) ~=~ 30$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(7) ~=~ 55$ $\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $a(8) ~=~ 105$ $\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(9) ~=~ 165$

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

 $1\!$ $a(1) ~=~ 0$ $\text{p}\!$ $a(2) ~=~ 1$ $\text{p}_\text{p}\!$ $a(3) ~=~ 2$ $\text{p}^\text{p}\!$ $a(4) ~=~ 2$ $\text{p}_{\text{p}_\text{p}}\!$ $a(5) ~=~ 3$ $\text{p} \text{p}_\text{p}\!$ $a(6) ~=~ 2$ $\text{p}_{\text{p}^\text{p}}\!$ $a(7) ~=~ 3$ $\text{p}^{\text{p}_\text{p}}\!$ $a(8) ~=~ 3$ $\text{p}_\text{p}^\text{p}\!$ $a(9) ~=~ 2$ $\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(10) ~=~ 3$ $\text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(11) ~=~ 4$ $\text{p}^\text{p} \text{p}_\text{p}\!$ $a(12) ~=~ 2$ $\text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(13) ~=~ 3$ $\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(14) ~=~ 3$ $\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(15) ~=~ 3$ $\text{p}^{\text{p}^\text{p}}\!$ $a(16) ~=~ 3$ $\text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $a(17) ~=~ 4$ $\text{p} \text{p}_\text{p}^\text{p}\!$ $a(18) ~=~ 2$ $\text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $a(19) ~=~ 4$ $\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(20) ~=~ 3$ $\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(21) ~=~ 3$ $\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(22) ~=~ 4$ $\text{p}_{\text{p}_\text{p}^\text{p}}\!$ $a(23) ~=~ 3$ $\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!$ $a(24) ~=~ 3$ $\text{p}_{\text{p}_\text{p}}^\text{p}\!$ $a(25) ~=~ 3$ $\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(26) ~=~ 3$ $\text{p}_\text{p}^{\text{p}_\text{p}}\!$ $a(27) ~=~ 3$ $\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(28) ~=~ 3$ $\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(29) ~=~ 4$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(30) ~=~ 3$ $\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!$ $a(31) ~=~ 5$ $\text{p}^{\text{p}_{\text{p}_\text{p}}}\!$ $a(32) ~=~ 4$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(33) ~=~ 4$ $\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $a(34) ~=~ 4$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $a(35) ~=~ 3$ $\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!$ $a(36) ~=~ 2$ $\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!$ $a(37) ~=~ 3$ $\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $a(38) ~=~ 4$ $\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(39) ~=~ 3$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!$ $a(40) ~=~ 3$ $\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!$ $a(41) ~=~ 4$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(42) ~=~ 3$ $\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!$ $a(43) ~=~ 4$ $\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(44) ~=~ 4$ $\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(45) ~=~ 3$ $\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!$ $a(46) ~=~ 3$ $\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(47) ~=~ 4$ $\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!$ $a(48) ~=~ 3$ $\text{p}_{\text{p}^\text{p}}^\text{p}\!$ $a(49) ~=~ 3$ $\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!$ $a(50) ~=~ 3$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $a(51) ~=~ 4$ $\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(52) ~=~ 3$ $\text{p}_{\text{p}^{\text{p}^\text{p}}}\!$ $a(53) ~=~ 4$ $\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!$ $a(54) ~=~ 3$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(55) ~=~ 4$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $a(56) ~=~ 3$ $\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $a(57) ~=~ 4$ $\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(58) ~=~ 4$ $\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!$ $a(59) ~=~ 5$ $\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(60) ~=~ 3$

## Miscellaneous Examples

$\text{Integers, Riffs, Rotes}\!$
 $\text{Integer}\!$ $\text{Riff}\!$ $\text{Rote}\!$
 $1\!$ $2\!$ $3\!$ $4\!$ $2010\!$ $2011\!$ $2012\!$ $2500\!$ $802701\!$ $123456789\!$