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

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

Rotes in Numerical Order

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

Prime Animations

Riffs 1 to 60

Rotes 1 to 60

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.

OEIS Entry for A061396.

{\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.

OEIS Entry for A062504.

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

{\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.

OEIS Entry for A062537.

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

A062860

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

OEIS Entry for A062860.

$a(n)={\text{Least Integer}}~j~{\text{with}}~n~{\text{Nodes in Its Riff}}$
$1\!$ $a(0)~=~1$	${\text{p}}\!$ $a(1)~=~2$	${\text{p}}_{\text{p}}\!$ $a(2)~=~3$	${\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(3)~=~5$	${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(4)~=~10$
${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(5)~=~15$	${\text{p}}{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(6)~=~30$	${\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(7)~=~55$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(8)~=~105$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(9)~=~165$

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.

OEIS Entry for A109301.

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.}}\!

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