Directory:Jon Awbrey/Papers/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 $n.$

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:

$\begin{matrix} 9876543210 & = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721 & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1 \end{matrix}$

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

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

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.

• OEIS Wiki 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 \'"`UNIQ-MathJax1-QINU`"' '"`UNIQ-MathJax2-QINU`"' '"`UNIQ-MathJax3-QINU`"' '"`UNIQ-MathJax4-QINU`"' :{| border="1" cellpadding="20" | [[Image:Rote 802701 Big.jpg|330px]] |} '"`UNIQ-MathJax5-QINU`"' <br> {| align="center" border="1" cellpadding="6" |+ style="height:25px" | \(a(n) = \text{Rote Height of}~ n$ $1\!$ $a(1) ~=~ 0$ $\text{p}\!$ $a(2) ~=~ 1$ $\text{p}_\text{p}\!$ $a(3) ~=~ 2$ $\text{p}^\text{p}\!$ $a(4) ~=~ 2$ $\text{p}_{\text{p}_\text{p}}\!$ $a(5) ~=~ 3$ $\text{p} \text{p}_\text{p}\!$ $a(6) ~=~ 2$ $\text{p}_{\text{p}^\text{p}}\!$ $a(7) ~=~ 3$ $\text{p}^{\text{p}_\text{p}}\!$ $a(8) ~=~ 3$ $\text{p}_\text{p}^\text{p}\!$ $a(9) ~=~ 2$ $\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(10) ~=~ 3$ $\text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(11) ~=~ 4$ $\text{p}^\text{p} \text{p}_\text{p}\!$ $a(12) ~=~ 2$ $\text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(13) ~=~ 3$ $\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(14) ~=~ 3$ $\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(15) ~=~ 3$ $\text{p}^{\text{p}^\text{p}}\!$ $a(16) ~=~ 3$ $\text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $a(17) ~=~ 4$ $\text{p} \text{p}_\text{p}^\text{p}\!$ $a(18) ~=~ 2$ $\text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $a(19) ~=~ 4$ $\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(20) ~=~ 3$ $\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(21) ~=~ 3$ $\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(22) ~=~ 4$ $\text{p}_{\text{p}_\text{p}^\text{p}}\!$ $a(23) ~=~ 3$ $\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!$ $a(24) ~=~ 3$ $\text{p}_{\text{p}_\text{p}}^\text{p}\!$ $a(25) ~=~ 3$ $\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(26) ~=~ 3$ $\text{p}_\text{p}^{\text{p}_\text{p}}\!$ $a(27) ~=~ 3$ $\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(28) ~=~ 3$ $\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(29) ~=~ 4$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(30) ~=~ 3$ $\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!$ $a(31) ~=~ 5$ $\text{p}^{\text{p}_{\text{p}_\text{p}}}\!$ $a(32) ~=~ 4$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(33) ~=~ 4$ $\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $a(34) ~=~ 4$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $a(35) ~=~ 3$ $\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!$ $a(36) ~=~ 2$ $\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!$ $a(37) ~=~ 3$ $\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $a(38) ~=~ 4$ $\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(39) ~=~ 3$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!$ $a(40) ~=~ 3$ $\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!$ $a(41) ~=~ 4$ $\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$ $a(42) ~=~ 3$ $\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!$ $a(43) ~=~ 4$ $\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(44) ~=~ 4$ $\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(45) ~=~ 3$ $\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!$ $a(46) ~=~ 3$ $\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(47) ~=~ 4$ $\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!$ $a(48) ~=~ 3$ $\text{p}_{\text{p}^\text{p}}^\text{p}\!$ $a(49) ~=~ 3$ $\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!$ $a(50) ~=~ 3$ $\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$ $a(51) ~=~ 4$ $\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$ $a(52) ~=~ 3$ $\text{p}_{\text{p}^{\text{p}^\text{p}}}\!$ $a(53) ~=~ 4$ $\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!$ $a(54) ~=~ 3$ $\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(55) ~=~ 4$ $\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $a(56) ~=~ 3$ $\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$ $a(57) ~=~ 4$ $\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$ $a(58) ~=~ 4$ $\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!$ $a(59) ~=~ 5$ $\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(60) ~=~ 3$