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# Riffs and Rotes

## Idea

Let ${\displaystyle {\text{p}}_{i}\!}$ be the ${\displaystyle i^{\text{th}}\!}$ prime, where the positive integer ${\displaystyle i\!}$ is called the index of the prime ${\displaystyle {\text{p}}_{i}\!}$ and the indices are taken in such a way that ${\displaystyle {\text{p}}_{1}=2.\!}$ Thus the sequence of primes begins as follows:

 ${\displaystyle {\begin{matrix}{\text{p}}_{1}=2,&{\text{p}}_{2}=3,&{\text{p}}_{3}=5,&{\text{p}}_{4}=7,&{\text{p}}_{5}=11,&{\text{p}}_{6}=13,&{\text{p}}_{7}=17,&{\text{p}}_{8}=19,&\ldots \end{matrix}}}$

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

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

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

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

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

For example:

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

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

Continuing with the same example, the index ${\displaystyle 504\!}$ has the factorization ${\displaystyle 2^{3}\cdot 3^{2}\cdot 7={\text{p}}_{1}^{3}{\text{p}}_{2}^{2}{\text{p}}_{4}^{1}\!}$ and the index ${\displaystyle 529\!}$ has the factorization ${\displaystyle {23}^{2}={\text{p}}_{9}^{2}.\!}$ Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:

 ${\displaystyle {\begin{array}{rcl}2&\mapsto &{\text{p}}_{1}^{1}\\[6pt]504&\mapsto &{\text{p}}_{1}^{3}{\text{p}}_{2}^{2}{\text{p}}_{4}^{1}\\[6pt]529&\mapsto &{\text{p}}_{9}^{2}\end{array}}}$

This leads to the following development:

 ${\displaystyle {\begin{array}{lll}123456789&=&{\text{p}}_{2}^{2}{\text{p}}_{504}^{1}{\text{p}}_{529}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}{\text{p}}_{{\text{p}}_{1}^{3}{\text{p}}_{2}^{2}{\text{p}}_{4}^{1}}^{1}{\text{p}}_{{\text{p}}_{9}^{2}}^{1}\end{array}}}$

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

 ${\displaystyle {\begin{array}{lll}123456789&=&{\text{p}}_{2}^{2}{\text{p}}_{504}^{1}{\text{p}}_{529}^{1}\\[18pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}{\text{p}}_{{\text{p}}_{1}^{3}{\text{p}}_{2}^{2}{\text{p}}_{4}^{1}}^{1}{\text{p}}_{{\text{p}}_{9}^{2}}^{1}\\[18pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{2}^{1}}{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}{\text{p}}_{{\text{p}}_{1}^{2}}^{1}}^{1}{\text{p}}_{{\text{p}}_{{\text{p}}_{2}^{2}}^{{\text{p}}_{1}^{1}}}^{1}\\[18pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}}^{1}{\text{p}}_{{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}}$

The ${\displaystyle 1\!}$'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:

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

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

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

 ${\displaystyle \operatorname {riff} (123456789)}$ is the following digraph: ${\displaystyle \operatorname {rote} (123456789)}$ is the following graph:

## Riffs in Numerical Order

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

## Rotes in Numerical Order

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

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

### A062504

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

### A062537

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

### A062860

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

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

 ${\displaystyle 1\!}$ ${\displaystyle a(1)~=~0}$