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{{DISPLAYTITLE:Riffs and Rotes}}
{{DISPLAYTITLE:Riffs and Rotes}}
__TOC__
<div class="nonumtoc">__TOC__</div>
==Riffs in Numerical Order==
==Idea==
Let <math>\text{p}_i\!</math> be the <math>i^\text{th}\!</math> prime, where the positive integer <math>i\!</math> is called the ''index'' of the prime <math>\text{p}_i\!</math> and the indices are taken in such a way that <math>\text{p}_1 = 2.\!</math> Thus the sequence of primes begins as follows:
{| align="center" cellpadding="6" width="90%"
|
<math>\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}</math>
|}
The prime factorization of a positive integer <math>n\!</math> can be written in the following form:
{| align="center" cellpadding="6" width="90%"
| <math>n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},\!</math>
|}
where <math>\text{p}_{i(k)}^{j(k)}\!</math> is the <math>k^\text{th}\!</math> prime power in the factorization and <math>\ell\!</math> is the number of distinct prime factors dividing <math>n.\!</math> The factorization of <math>1\!</math> is defined as <math>1\!</math> in accord with the convention that an empty product is equal to <math>1.\!</math>
Let <math>I(n)\!</math> be the set of indices of primes that divide <math>n\!</math> and let <math>j(i, n)\!</math> be the number of times that <math>\text{p}_i\!</math> divides <math>n.\!</math> Then the prime factorization of <math>n\!</math> can be written in the following alternative form:
{| align="center" cellpadding="6" width="90%"
| <math>n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.\!</math>
|}
For example:
{| align="center" border="1" cellpadding="10"
{| align="center" cellpadding="6" width="90%"
|+ style="height:25px" | <math>\text{Riffs in Numerical Order}\!</math>
|
| valign="bottom" |
<math>\begin{matrix}
<p> </p><br>
123456789
<p><math>1\!</math></p><br>
& = & 3^2 \cdot 3607 \cdot 3803
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1.
\end{matrix}</math>
|}
Each index <math>i\!</math> and exponent <math>j\!</math> appearing in the prime factorization of a positive integer <math>n\!</math> is itself a positive integer, and thus has a prime factorization of its own.
| valign="bottom" |
<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
Continuing with the same example, the index <math>504\!</math> has the factorization <math>2^3 \cdot 3^2 \cdot 7 = \text{p}_1^3 \text{p}_2^2 \text{p}_4^1\!</math> and the index <math>529\!</math> has the factorization <math>{23}^2 = \text{p}_9^2.\!</math> Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:
{| align="center" cellpadding="6" width="90%"
| valign="bottom" |
|
<p>[[Image:Riff 4 Big.jpg|40px]]</p><br>
<math>\begin{array}{rcl}
2 & \mapsto & \text{p}_1^1
\\[6pt]
| valign="bottom" |
504 & \mapsto & \text{p}_1^3 \text{p}_2^2 \text{p}_4^1
<p>[[Image:Riff 5 Big.jpg|65px]]</p><br>
\\[6pt]
529 & \mapsto & \text{p}_9^2
<p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
\end{array}</math>
|}
| valign="bottom" |
<p>[[Image:Riff 6 Big.jpg|65px]]</p><br>
This leads to the following development:
{| align="center" cellpadding="6" width="90%"
|
<math>\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}</math>
|}
Continuing to replace every index and exponent with its factorization produces the following development:
{| align="center" cellpadding="6" width="90%"
|
<math>\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}</math>
|}
The <math>1\!</math>'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:
{| align="center" cellpadding="6" width="90%"
|
<math>\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}</math>
|}
The pattern of indices and exponents illustrated here is called a ''doubly recursive factorization'', or ''DRF''. Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the DRF of <math>n.\!</math> If <math>\mathbb{M}</math> is the set of positive integers, <math>\mathcal{L}</math> is the set of DRF expressions, and the mapping defined by the factorization process is denoted <math>\operatorname{drf} : \mathbb{M} \to \mathcal{L},</math> then the doubly recursive factorization of <math>n\!</math> is denoted <math>\operatorname{drf}(n).\!</math>
The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called ''riffs'' and ''rotes'', respectively.
{| align=center cellpadding="6" width="90%"
|-
| <math>\operatorname{riff}(123456789)</math> is the following digraph:
|-
| align=center | [[Image:Riff 123456789 Big.jpg|220px]]
|-
| <math>\operatorname{rote}(123456789)</math> is the following graph:
|-
| align=center | [[Image:Rote 123456789 Big.jpg|345px]]
|}
==Riffs in Numerical Order==
{| align="center" border="1" cellpadding="12"
|+ style="height:25px" | <math>\text{Riffs in Numerical Order}\!</math>
| valign="bottom" |
| valign="bottom" |
<p>[[Image:Riff 7 Big.jpg|65px]]</p><br>
<p> </p><br>
<p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>1\!</math></p><br>
<p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
| valign="bottom" |
<p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
<p><math>\text{p}\!</math></p><br>
<p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p>
| valign="bottom" |
<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
<p><math>\text{p}_\text{p}\!</math></p><br>
<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
| valign="bottom" |
<p>[[Image:Riff 4 Big.jpg|40px]]</p><br>
<p><math>\text{p}^\text{p}\!</math></p><br>
<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
| valign="bottom" |
<p>[[Image:Riff 5 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 6 Big.jpg|65px]]</p><br>
<p><math>\text{p} \text{p}_\text{p}\!</math></p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
| valign="bottom" |
<p>[[Image:Riff 7 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>\begin{array}{l} 4\!:\!1 \\ 7 \end{array}</math></p>
<p><math>\begin{array}{l} 4\!:\!1 \\ 7 \end{array}</math></p>
| valign="bottom" |
| valign="bottom" |
{| align="center" border="1" cellpadding="6"
{| align="center" border="1" cellpadding="6"
|+ style="height:25px" | <math>\text{Rotes in Numerical Order}\!</math>
| valign="bottom" |
| valign="bottom" |
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
|}
|}
==Selected Sequences==
==Prime Animations==
===Riffs 1 to 60===
{| align="center" border="1" width="90%"
{| align="center"
|+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math>
| [[Image:Animation Riff 60 x 0.16.gif]]
|}
===Rotes 1 to 60===
{| align="center"
| [[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.'''
* [http://oeis.org/A061396 OEIS Entry for A061396].
{| align="center" border="1" width="96%"
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
|- style="height:50px; background:#f0f0ff"
|- style="height:50px; background:#f0f0ff"
|
|
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
| width="10%" | <math>\text{Integer}\!</math>
| width="10%" | <math>\text{Integer}\!</math>
| width="25%" | <math>\text{Factorization}\!</math>
| width="19%" | <math>\text{Factorization}\!</math>
| width="15%" | <math>\text{Notation}\!</math>
| width="14%" | <math>\text{Notation}\!</math>
| width="25%" | <math>\text{Riff Digraph}\!</math>
| width="19%" | <math>\text{Riff Digraph}\!</math>
| width="25%" | <math>\text{Rote Graph}\!</math>
| width="19%" | <math>\text{Rote Graph}\!</math>
| width="19%" | <math>\text{Traversal}\!</math>
|}
|}
|-
|-
{| cellpadding="12" style="text-align:center; width:100%"
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>1\!</math>
| width="10%" | <math>1\!</math>
| width="25%" | <math>1\!</math>
| width="19%" | <math>1\!</math>
| width="15%" |
| width="14%" |
| width="25%" |
| width="19%" |
| width="25%" | [[Image:Rote 1 Big.jpg|20px]]
| width="19%" | [[Image:Rote 1 Big.jpg|20px]]
| width="19%" |
|}
|}
|-
|-
{| cellpadding="12" style="text-align:center; width:100%"
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>2\!</math>
| width="10%" | <math>2\!</math>
| width="25%" | <math>\text{p}_1^1\!</math>
| width="19%" | <math>\text{p}_1^1\!</math>
| width="15%" | <math>\text{p}\!</math>
| width="14%" | <math>\text{p}\!</math>
| width="25%" | [[Image:Riff 2 Big.jpg|20px]]
| width="19%" | [[Image:Riff 2 Big.jpg|20px]]
| width="25%" | [[Image:Rote 2 Big.jpg|40px]]
| width="19%" | [[Image:Rote 2 Big.jpg|40px]]
| width="19%" | <math>((~))</math>
|}
|}
|-
|-
{| cellpadding="12" style="text-align:center; width:100%"
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>3\!</math>
| width="10%" | <math>3\!</math>
| width="25%" |
| width="19%" |
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
\end{array}</math>
| width="15%" | <math>\text{p}_\text{p}\!</math>
| width="14%" | <math>\text{p}_\text{p}\!</math>
| width="25%" | [[Image:Riff 3 Big.jpg|40px]]
| width="19%" | [[Image:Riff 3 Big.jpg|40px]]
| width="25%" | [[Image:Rote 3 Big.jpg|40px]]
| width="19%" | [[Image:Rote 3 Big.jpg|40px]]
| width="19%" | <math>(((~))(~))</math>
|-
|-
| <math>4\!</math>
| <math>4\!</math>
| [[Image:Riff 4 Big.jpg|40px]]
| [[Image:Riff 4 Big.jpg|40px]]
| [[Image:Rote 4 Big.jpg|65px]]
| [[Image:Rote 4 Big.jpg|65px]]
| <math>((((~))))</math>
|}
|}
|-
|-
{| cellpadding="12" style="text-align:center; width:100%"
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>5\!</math>
| width="10%" | <math>5\!</math>
| width="25%" |
| width="19%" |
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_3^1
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
& = & \text{p}_{\text{p}_2^1}^1
\\[12pt]
\\[10pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
\end{array}</math>
| width="15%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="25%" | [[Image:Riff 5 Big.jpg|65px]]
| width="19%" | [[Image:Riff 5 Big.jpg|65px]]
| width="25%" | [[Image:Rote 5 Big.jpg|40px]]
| width="19%" | [[Image:Rote 5 Big.jpg|40px]]
| width="19%" | <math>((((~))(~))(~))</math>
|-
|-
| <math>6\!</math>
| <math>6\!</math>
| [[Image:Riff 6 Big.jpg|65px]]
| [[Image:Riff 6 Big.jpg|65px]]
| [[Image:Rote 6 Big.jpg|80px]]
| [[Image:Rote 6 Big.jpg|80px]]
| <math>((~))(((~))(~))</math>
|-
|-
| <math>7\!</math>
| <math>7\!</math>
\text{p}_4^1
\text{p}_4^1
& = & \text{p}_{\text{p}_1^2}^1
& = & \text{p}_{\text{p}_1^2}^1
\\[12pt]
\\[10pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}</math>
\end{array}</math>
| [[Image:Riff 7 Big.jpg|65px]]
| [[Image:Riff 7 Big.jpg|65px]]
| [[Image:Rote 7 Big.jpg|65px]]
| [[Image:Rote 7 Big.jpg|65px]]
| <math>(((((~))))(~))</math>
|-
|-
| <math>8\!</math>
| <math>8\!</math>
\text{p}_1^3
\text{p}_1^3
& = & \text{p}_1^{\text{p}_2^1}
& = & \text{p}_1^{\text{p}_2^1}
\\[12pt]
\\[10pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}</math>
\end{array}</math>
| [[Image:Riff 8 Big.jpg|65px]]
| [[Image:Riff 8 Big.jpg|65px]]
| [[Image:Rote 8 Big.jpg|65px]]
| [[Image:Rote 8 Big.jpg|65px]]
| <math>(((((~))(~))))</math>
|-
|-
| <math>9\!</math>
| <math>9\!</math>
| [[Image:Riff 9 Big.jpg|40px]]
| [[Image:Riff 9 Big.jpg|40px]]
| [[Image:Rote 9 Big.jpg|80px]]
| [[Image:Rote 9 Big.jpg|80px]]
| <math>(((~))(((~))))</math>
|-
|-
| <math>16\!</math>
| <math>16\!</math>
\text{p}_1^4
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
& = & \text{p}_1^{\text{p}_1^2}
\\[12pt]
\\[10pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}</math>
\end{array}</math>
| [[Image:Riff 16 Big.jpg|65px]]
| [[Image:Riff 16 Big.jpg|65px]]
| [[Image:Rote 16 Big.jpg|90px]]
| [[Image:Rote 16 Big.jpg|90px]]
| <math>((((((~))))))</math>
|}
|}
|}
===A062504===
* '''Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.'''
* [http://oeis.org/A062504 OEIS Entry for A062504].
{| align="center"
| width="25%" | [[Image:Rote 10 Big.jpg|80px]]
|
|
<math>\begin{array}{lll}
<math>\begin{array}{l|l|r}
\text{p}_5^1
k
& P_k
\\[12pt]
= \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \}
= \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \}
\\[12pt]
& |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}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math>
|}
| [[Image:Riff 11 Big.jpg|90px]]
{| align="center" border="1" width="90%"
|+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math>
|- style="height:50px; background:#f0f0ff"
|
|
<math>\begin{array}{lll}
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
\text{p}_1^2 \text{p}_2^1
| width="10%" | <math>\text{Integer}\!</math>
| width="25%" | <math>\text{Factorization}\!</math>
| width="15%" | <math>\text{Notation}\!</math>
| <math>\text{p}^{\text{p}} \text{p}_{\text{p}}\!</math>
| width="25%" | <math>\text{Riff Digraph}\!</math>
| [[Image:Riff 12 Big.jpg|65px]]
| width="25%" | <math>\text{Rote Graph}\!</math>
|}
|-
|-
|
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>1\!</math>
| width="25%" | <math>1\!</math>
| width="15%" |
| width="25%" |
| width="25%" | [[Image:Rote 1 Big.jpg|20px]]
| <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math>
|}
| [[Image:Riff 13 Big.jpg|65px]]
| [[Image:Rote 13 Big.jpg|80px]]
|-
|-
|
|
<math>\begin{array}{lll}
{| cellpadding="12" style="text-align:center; width:100%"
\text{p}_1^1 \text{p}_4^1
| width="10%" | <math>2\!</math>
| width="25%" | <math>\text{p}_1^1\!</math>
| width="15%" | <math>\text{p}\!</math>
& = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
| width="25%" | [[Image:Riff 2 Big.jpg|20px]]
| width="25%" | [[Image:Rote 2 Big.jpg|40px]]
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>3\!</math>
| width="25%" |
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math>
| width="15%" | <math>\text{p}_\text{p}\!</math>
| [[Image:Riff 14 Big.jpg|90px]]
| width="25%" | [[Image:Riff 3 Big.jpg|40px]]
| [[Image:Rote 14 Big.jpg|105px]]
| width="25%" | [[Image:Rote 3 Big.jpg|40px]]
|-
|-
| <math>17\!</math>
| <math>4\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math>
| <math>\text{p}^\text{p}\!</math>
| [[Image:Riff 17 Big.jpg|90px]]
| [[Image:Riff 4 Big.jpg|40px]]
| [[Image:Rote 17 Big.jpg|65px]]
| [[Image:Rote 4 Big.jpg|65px]]
|}
|-
|-
|
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>5\!</math>
| width="25%" |
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^2
\text{p}_3^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
& = & \text{p}_{\text{p}_2^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math>
| width="15%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| [[Image:Riff 18 Big.jpg|65px]]
| width="25%" | [[Image:Riff 5 Big.jpg|65px]]
| [[Image:Rote 18 Big.jpg|120px]]
| width="25%" | [[Image:Rote 5 Big.jpg|40px]]
|-
|-
| <math>19\!</math>
| <math>6\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
& = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| [[Image:Riff 19 Big.jpg|90px]]
| [[Image:Riff 6 Big.jpg|65px]]
| [[Image:Rote 19 Big.jpg|65px]]
| [[Image:Rote 6 Big.jpg|80px]]
|-
|-
| <math>23\!</math>
| <math>7\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_9^1
\text{p}_4^1
& = & \text{p}_{\text{p}_2^2}^1
& = & \text{p}_{\text{p}_1^2}^1
\\[12pt]
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 23 Big.jpg|65px]]
| [[Image:Riff 7 Big.jpg|65px]]
| [[Image:Rote 23 Big.jpg|80px]]
| [[Image:Rote 7 Big.jpg|65px]]
|-
|-
| <math>25\!</math>
| <math>8\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_3^2
\text{p}_1^3
& = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1}
& = & \text{p}_1^{\text{p}_2^1}
\\[12pt]
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1}
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| [[Image:Riff 25 Big.jpg|65px]]
| [[Image:Riff 8 Big.jpg|65px]]
| [[Image:Rote 25 Big.jpg|80px]]
| [[Image:Rote 8 Big.jpg|65px]]
|-
|-
| <math>27\!</math>
| <math>9\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_2^3
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1}
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| [[Image:Riff 27 Big.jpg|65px]]
| [[Image:Riff 9 Big.jpg|40px]]
| [[Image:Rote 27 Big.jpg|80px]]
| [[Image:Rote 9 Big.jpg|80px]]
|-
|-
| <math>32\!</math>
| <math>16\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^5
\text{p}_1^4
& = & \text{p}_1^{\text{p}_3^1}
& = & \text{p}_1^{\text{p}_1^2}
\\[12pt]
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1}
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 32 Big.jpg|90px]]
| [[Image:Riff 16 Big.jpg|65px]]
| [[Image:Rote 32 Big.jpg|65px]]
| [[Image:Rote 16 Big.jpg|90px]]
|}
|-
|-
|
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>10\!</math>
| width="25%" |
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_4^2
\text{p}_1^1 \text{p}_3^1
& = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1}
& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1
\\[12pt]
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1}
& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math>
| width="15%" | <math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math>
| [[Image:Riff 49 Big.jpg|65px]]
| width="25%" | [[Image:Riff 10 Big.jpg|90px]]
| [[Image:Rote 49 Big.jpg|80px]]
| width="25%" | [[Image:Rote 10 Big.jpg|80px]]
|-
|-
| <math>53\!</math>
| <math>11\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_{16}^1
\text{p}_5^1
& = & \text{p}_{\text{p}_1^4}^1
& = & \text{p}_{\text{p}_3^1}^1
\\[12pt]
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1
& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1
\\[12pt]
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1
& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math>
| <math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math>
| [[Image:Riff 53 Big.jpg|90px]]
| [[Image:Riff 11 Big.jpg|90px]]
| [[Image:Rote 53 Big.jpg|90px]]
| [[Image:Rote 11 Big.jpg|40px]]
|-
|-
| <math>64\!</math>
| <math>12\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^6
\text{p}_1^2 \text{p}_2^1
& = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1
& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
| <math>\text{p}^{\text{p}} \text{p}_{\text{p}}\!</math>
| [[Image:Riff 64 Big.jpg|65px]]
| [[Image:Riff 12 Big.jpg|65px]]
| [[Image:Rote 64 Big.jpg|105px]]
| [[Image:Rote 12 Big.jpg|105px]]
|-
|-
| <math>81\!</math>
| <math>13\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_2^4
\text{p}_6^1
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2}
& = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1
\\[12pt]
\\[12pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}}
& = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
| <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math>
| [[Image:Riff 81 Big.jpg|65px]]
| [[Image:Riff 13 Big.jpg|65px]]
| [[Image:Rote 81 Big.jpg|105px]]
| [[Image:Rote 13 Big.jpg|80px]]
|-
|-
| <math>128\!</math>
| <math>14\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^7
\text{p}_1^1 \text{p}_4^1
& = & \text{p}_1^{\text{p}_4^1}
& = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1
\\[12pt]
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1}
& = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
| <math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 128 Big.jpg|90px]]
| [[Image:Riff 14 Big.jpg|90px]]
| [[Image:Rote 128 Big.jpg|90px]]
| [[Image:Rote 14 Big.jpg|105px]]
|-
|-
| <math>256\!</math>
| <math>17\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^8
\text{p}_7^1
& = & \text{p}_1^{\text{p}_1^3}
& = & \text{p}_{\text{p}_4^1}^1
\\[12pt]
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}}
& = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1
\\[12pt]
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}
& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math>
| <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 256 Big.jpg|90px]]
| [[Image:Riff 17 Big.jpg|90px]]
| [[Image:Rote 256 Big.jpg|90px]]
| [[Image:Rote 17 Big.jpg|65px]]
|-
|-
| <math>512\!</math>
| <math>18\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^9
\text{p}_1^1 \text{p}_2^2
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math>
| <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math>
| [[Image:Riff 512 Big.jpg|65px]]
| [[Image:Riff 18 Big.jpg|65px]]
| [[Image:Rote 512 Big.jpg|105px]]
| [[Image:Rote 18 Big.jpg|120px]]
|-
|-
| <math>65536\!</math>
| <math>19\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^{16}
\text{p}_8^1
& = & \text{p}_1^{\text{p}_1^4}
& = & \text{p}_{\text{p}_1^3}^1
\\[12pt]
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}}
& = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1
\\[12pt]
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}
& = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
| <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math>
| [[Image:Riff 65536 Big.jpg|90px]]
| [[Image:Riff 19 Big.jpg|90px]]
| [[Image:Rote 65536 Big.jpg|115px]]
| [[Image:Rote 19 Big.jpg|65px]]
|-
|-
| <math>23\!</math>
|
|
<math>\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}</math>
| <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math>
| [[Image:Riff 23 Big.jpg|65px]]
| [[Image:Rote 23 Big.jpg|80px]]
{| cellpadding="12" style="text-align:center; width:100%"
|-
|-
| <math>25\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\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}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math>
| [[Image:Riff 25 Big.jpg|65px]]
| [[Image:Rote 25 Big.jpg|80px]]
|-
|-
| <math>4\!</math>
| <math>27\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\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}</math>
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math>
| [[Image:Riff 4 Big.jpg|40px]]
| [[Image:Riff 27 Big.jpg|65px]]
| [[Image:Rote 4 Big.jpg|65px]]
| [[Image:Rote 27 Big.jpg|80px]]
|-
|-
| <math>32\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_1^5
& = & \text{p}_1^{\text{p}_3^1}
<math>\begin{array}{lll}
\\[12pt]
\text{p}_3^1
& = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1}
& = & \text{p}_{\text{p}_2^1}^1
\\[12pt]
\\[10pt]
& = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math>
| [[Image:Riff 32 Big.jpg|90px]]
| [[Image:Rote 32 Big.jpg|65px]]
|-
|-
| <math>6\!</math>
| <math>49\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
\text{p}_4^2
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
& = & \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}</math>
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math>
| [[Image:Riff 6 Big.jpg|65px]]
| [[Image:Riff 49 Big.jpg|65px]]
| [[Image:Rote 6 Big.jpg|80px]]
| [[Image:Rote 49 Big.jpg|80px]]
|-
|-
| <math>7\!</math>
| <math>53\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_4^1
\text{p}_{16}^1
& = & \text{p}_{\text{p}_1^2}^1
& = & \text{p}_{\text{p}_1^4}^1
\\[10pt]
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
& = & \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}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 7 Big.jpg|65px]]
| [[Image:Riff 53 Big.jpg|90px]]
| [[Image:Rote 7 Big.jpg|65px]]
| [[Image:Rote 53 Big.jpg|90px]]
|-
|-
| <math>8\!</math>
| <math>64\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^3
\text{p}_1^6
& = & \text{p}_1^{\text{p}_2^1}
& = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1}
\\[10pt]
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
| [[Image:Riff 8 Big.jpg|65px]]
| [[Image:Riff 64 Big.jpg|65px]]
| [[Image:Rote 8 Big.jpg|65px]]
| [[Image:Rote 64 Big.jpg|105px]]
|-
|-
| <math>9\!</math>
| <math>81\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_2^2
\text{p}_2^4
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
& = & \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}</math>
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 9 Big.jpg|40px]]
| [[Image:Riff 81 Big.jpg|65px]]
| [[Image:Rote 9 Big.jpg|80px]]
| [[Image:Rote 81 Big.jpg|105px]]
|-
|-
| <math>16\!</math>
| <math>128\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_1^4
\text{p}_1^7
& = & \text{p}_1^{\text{p}_1^2}
& = & \text{p}_1^{\text{p}_4^1}
\\[10pt]
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
& = & \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}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 16 Big.jpg|65px]]
| [[Image:Riff 128 Big.jpg|90px]]
| [[Image:Rote 16 Big.jpg|90px]]
| [[Image:Rote 128 Big.jpg|90px]]
| <math>((((((~))))))</math>
|-
|}
| <math>256\!</math>
|
<math>\begin{array}{lll}
===A062504===
\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}</math>
| <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math>
| [[Image:Riff 256 Big.jpg|90px]]
| [[Image:Rote 256 Big.jpg|90px]]
|-
| <math>512\!</math>
|
|
<math>\begin{array}{l|l|r}
<math>\begin{array}{lll}
\text{p}_1^9
& P_k
& = & \text{p}_1^{\text{p}_2^2}
= \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math>
| [[Image:Riff 512 Big.jpg|65px]]
\\
| [[Image:Rote 512 Big.jpg|105px]]
|-
\\
| <math>65536\!</math>
|
\\
<math>\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}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 65536 Big.jpg|90px]]
| [[Image:Rote 65536 Big.jpg|115px]]
|}
|}
|}
* '''Nodes in riff (rooted index-functional forest) for n.'''
* '''Nodes in riff (rooted index-functional forest) for n.'''
* [http://oeis.org/wiki/A062537 OEIS Wiki Entry for A062537].
* [http://oeis.org/A062537 OEIS Entry for A062537].
{| align="center" border="1" cellpadding="10"
{| align="center" border="1" cellpadding="10"
* '''Smallest j with n nodes in its riff (rooted index-functional forest).'''
* '''Smallest j with n nodes in its riff (rooted index-functional forest).'''
* [http://oeis.org/wiki/A062860 OEIS Wiki Entry for A062860].
* [http://oeis.org/A062860 OEIS Entry for A062860].
{| align="center" border="1" cellpadding="10"
{| align="center" border="1" cellpadding="10"
* '''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.'''
* '''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.'''
* [http://oeis.org/wiki/A109301 OEIS Wiki Entry for A109301].
* [http://oeis.org/A109301 OEIS Entry for A109301].
; Example
: <math>802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1</math>
: <math>\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}</math>
: <math>\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}</math>
: <math>\text{So the rote of 802701 is the following graph:}\!</math>
:{| border="1" cellpadding="20"
| [[Image:Rote 802701 Big.jpg|330px]]
|}
: <math>\text{By inspection, the rote height of 802701 is 6.}\!</math>
<br>
{| align="center" border="1" cellpadding="6"
{| align="center" border="1" cellpadding="6"
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(60) ~=~ 3</math></p>
<p><math>a(60) ~=~ 3</math></p>
|}
==Miscellaneous Examples==
{| align="center" border="1" width="96%"
|+ style="height:24px" | <math>\text{Integers, Riffs, Rotes}\!</math>
|- style="height:50px; background:#f0f0ff"
|
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
| width="10%" | <math>\text{Integer}\!</math>
| width="45%" | <math>\text{Riff}\!</math>
| width="45%" | <math>\text{Rote}\!</math>
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>1\!</math>
| width="45%" |
| width="45%" | [[Image:Rote 1 Big.jpg|15px]]
|-
| <math>2\!</math>
| [[Image:Riff 2 Big.jpg|15px]]
| [[Image:Rote 2 Big.jpg|30px]]
|-
| <math>3\!</math>
| [[Image:Riff 3 Big.jpg|30px]]
| [[Image:Rote 3 Big.jpg|30px]]
|-
| <math>4\!</math>
| [[Image:Riff 4 Big.jpg|30px]]
| [[Image:Rote 4 Big.jpg|48px]]
|-
| <math>360\!</math>
| [[Image:Riff 360 Big.jpg|120px]]
| [[Image:Rote 360 Big.jpg|135px]]
|-
| <math>2010\!</math>
| [[Image:Riff 2010 Big.jpg|138px]]
| [[Image:Rote 2010 Big.jpg|144px]]
|-
| <math>2011\!</math>
| [[Image:Riff 2011 Big.jpg|84px]]
| [[Image:Rote 2011 Big.jpg|120px]]
|-
| <math>2012\!</math>
| [[Image:Riff 2012 Big.jpg|100px]]
| [[Image:Rote 2012 Big.jpg|125px]]
|-
| <math>2500\!</math>
| [[Image:Riff 2500 Big.jpg|66px]]
| [[Image:Rote 2500 Big.jpg|125px]]
|-
| <math>802701\!</math>
| [[Image:Riff 802701 Big.jpg|156px]]
| [[Image:Rote 802701 Big.jpg|245px]]
|-
| <math>123456789\!</math>
| [[Image:Riff 123456789 Big.jpg|162px]]
| [[Image:Rote 123456789 Big.jpg|256px]]
|}
|}
|}
