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Riffs and Rotes

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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


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

Image:Animation Riff 60 x 0.16.gif

Rotes 1 to 60

Image:Animation Rote 60 x 0.16.gif

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.
\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.

\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.
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).
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.
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

Miscellaneous Examples

\text{Integers, Riffs, Rotes}\!
\text{Integer}\! \text{Riff}\! \text{Rote}\!
1\!  
2\!
3\!
4\!
2010\!
2011\!
2012\!
2500\!
802701\!
123456789\!
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