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:

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

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.

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