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 above expression:

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