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MyWikiBiz, Author Your Legacy — Friday October 31, 2025
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Revision as of 18:48, 31 January 2010
, 18:48, 31 January 2010Undo revision 107256 by Jon Awbrey (Talk)
===Plain Wiki Table===
===Plain Wiki Table===
====Large Scale====
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
|+ style="height:24px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
|- style="height:48px; background:#f0f0ff"
|- style="height:48px; background:#f0f0ff"
| <math>\text{Integer}\!</math>
| <math>\text{Integer}\!</math>
|
|
|
|
| [[Image:Rooted Node Big.jpg|20px]]
| [[Image:Rote 1 Big.jpg|20px]]
|
|
|-
|-
| <math>\text{p}_1^1\!</math>
| <math>\text{p}_1^1\!</math>
| <math>\text{p}\!</math>
| <math>\text{p}\!</math>
| [[Image:Rooted Node Big.jpg|20px]]
| [[Image:Riff 2 Big.jpg|20px]]
| [[Image:Rote 2 Big.jpg|40px]]
| [[Image:Rote 2 Big.jpg|40px]]
| <math>((~))</math>
| <math>((~))</math>
\end{array}</math>
\end{array}</math>
| <math>\text{p}_\text{p}\!</math>
| <math>\text{p}_\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Riff 3 Big.jpg|40px]]
| [[Image:Rote 3 Big.jpg|40px]]
| [[Image:Rote 3 Big.jpg|40px]]
| <math>(((~))(~))</math>
| <math>(((~))(~))</math>
\end{array}</math>
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| <math>\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Riff 4 Big.jpg|40px]]
| [[Image:Rote 4 Big.jpg|65px]]
| [[Image:Rote 4 Big.jpg|65px]]
| <math>((((~))))</math>
| <math>((((~))))</math>
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 5 Big.jpg|65px]]
| [[Image:Rote 5 Big.jpg|40px]]
| [[Image:Rote 5 Big.jpg|40px]]
| <math>((((~))(~))(~))</math>
| <math>((((~))(~))(~))</math>
\end{array}</math>
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 6 Big.jpg|65px]]
| [[Image:Rote 6 Big.jpg|80px]]
| [[Image:Rote 6 Big.jpg|80px]]
| <math>((~))(((~))(~))</math>
| <math>((~))(((~))(~))</math>
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 7 Big.jpg|65px]]
| [[Image:Rote 7 Big.jpg|65px]]
| [[Image:Rote 7 Big.jpg|65px]]
| <math>(((((~))))(~))</math>
| <math>(((((~))))(~))</math>
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 8 Big.jpg|65px]]
| [[Image:Rote 8 Big.jpg|65px]]
| [[Image:Rote 8 Big.jpg|65px]]
| <math>(((((~))(~))))</math>
| <math>(((((~))(~))))</math>
\end{array}</math>
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Riff 9 Big.jpg|40px]]
| [[Image:Rote 9 Big.jpg|80px]]
| [[Image:Rote 9 Big.jpg|80px]]
| <math>(((~))(((~))))</math>
| <math>(((~))(((~))))</math>
\end{array}</math>
\end{array}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 16 Big.jpg|65px]]
| [[Image:Rote 16 Big.jpg|90px]]
| [[Image:Rote 16 Big.jpg|90px]]
| <math>((((((~))))))</math>
| <math>((((((~))))))</math>
|}
|}
====Small Scale====
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
|- style="height:48px; background:#f0f0ff"
| <math>\text{Integer}\!</math>
{| align="center" border="1" width="96%"
| <math>\text{Factorization}\!</math>
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
| <math>\text{Notation}\!</math>
|- style="height:50px; background:#f0f0ff"
| <math>\text{Riff Digraph}\!</math>
| <math>\text{Rote Graph}\!</math>
| <math>\text{Traversal}\!</math>
|- style="height:48px"
| <math>1\!</math>
| <math>1\!</math>
|
|
| [[Image:Rote 1 Big.jpg|12px]]
|
|-
| <math>2\!</math>
| <math>\text{p}_1^1\!</math>
| <math>\text{p}\!</math>
| [[Image:Riff 2 Big.jpg|12px]]
| [[Image:Rote 2 Big.jpg|24px]]
| <math>((~))</math>
|-
| <math>3\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p}_\text{p}\!</math>
| [[Image:Riff 3 Big.jpg|24px]]
| [[Image:Rote 3 Big.jpg|24px]]
| width="19%" | <math>\text{Traversal}\!</math>
| <math>(((~))(~))</math>
|-
|-
| <math>4\!</math>
|
|
{| cellpadding="12" style="text-align:center; width:100%"
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}</math>
| width="14%" |
| <math>\text{p}^\text{p}\!</math>
| [[Image:Riff 4 Big.jpg|24px]]
| [[Image:Rote 4 Big.jpg|38px]]
| width="19%" |
| <math>((((~))))</math>
|-
|-
| <math>5\!</math>
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\\[6pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="19%" | <math>\cdots</math>
| [[Image:Riff 5 Big.jpg|38px]]
| [[Image:Rote 5 Big.jpg|24px]]
| <math>((((~))(~))(~))</math>
|-
|-
| <math>4\!</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}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^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}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 6 Big.jpg|38px]]
| [[Image:Rote 6 Big.jpg|80px]]
| [[Image:Rote 6 Big.jpg|48px]]
| <math>((~))(((~))(~))</math>
| <math>((~))(((~))(~))</math>
|-
|-
\text{p}_4^1
\text{p}_4^1
& = & \text{p}_{\text{p}_1^2}^1
& = & \text{p}_{\text{p}_1^2}^1
\\[10pt]
\\[6pt]
& = & \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>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 7 Big.jpg|38px]]
| [[Image:Rote 7 Big.jpg|65px]]
| [[Image:Rote 7 Big.jpg|38px]]
| <math>(((((~))))(~))</math>
| <math>(((((~))))(~))</math>
|-
|-
\text{p}_1^3
\text{p}_1^3
& = & \text{p}_1^{\text{p}_2^1}
& = & \text{p}_1^{\text{p}_2^1}
\\[10pt]
\\[6pt]
& = & \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>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 8 Big.jpg|38px]]
| [[Image:Rote 8 Big.jpg|65px]]
| [[Image:Rote 8 Big.jpg|38px]]
| <math>(((((~))(~))))</math>
| <math>(((((~))(~))))</math>
|-
|-
\end{array}</math>
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Riff 9 Big.jpg|24px]]
| [[Image:Rote 9 Big.jpg|80px]]
| [[Image:Rote 9 Big.jpg|48px]]
| <math>(((~))(((~))))</math>
| <math>(((~))(((~))))</math>
|-
|-
\text{p}_1^4
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
& = & \text{p}_1^{\text{p}_1^2}
\\[10pt]
\\[6pt]
& = & \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>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Riff 16 Big.jpg|38px]]
| [[Image:Rote 16 Big.jpg|90px]]
| [[Image:Rote 16 Big.jpg|52px]]
| <math>((((((~))))))</math>
| <math>((((((~))))))</math>
|}
|}
===Nested Wiki Table===
===Old ASCII Version===
====Large Scale====
{| align="center" border="1" width="96%"
| integer factorization riff r.i.f.f. rote --> in parentheses
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
| k p's k nodes 2k+1 nodes
|- style="height:50px; background:#f0f0ff"
|
|
| 1 1 blank blank @ blank
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
| width="10%" | <math>\text{Integer}\!</math>
| width="19%" | <math>\text{Factorization}\!</math>
| width="14%" | <math>\text{Notation}\!</math>
| width="19%" | <math>\text{Riff Digraph}\!</math>
| width="19%" | <math>\text{Rote Graph}\!</math>
| width="19%" | <math>\text{Traversal}\!</math>
|}
|-
|
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>1\!</math>
| width="19%" | <math>1\!</math>
| width="14%" |
| width="19%" |
| width="19%" | [[Image:Rote 1 Big.jpg|20px]]
| width="19%" |
|}
|-
|
|
| o---o
{| cellpadding="12" style="text-align:center; width:100%"
| |
| width="10%" | <math>2\!</math>
| 2 p_1^1 p @ @ (())
| width="19%" | <math>\text{p}_1^1\!</math>
|
| width="14%" | <math>\text{p}\!</math>
| width="19%" | [[Image:Riff 2 Big.jpg|20px]]
| width="19%" | [[Image:Rote 2 Big.jpg|40px]]
| width="19%" | <math>((~))</math>
|}
|-
|
|
| o---o
{| cellpadding="12" style="text-align:center; width:100%"
| |
| width="10%" | <math>3\!</math>
| o---o
| width="19%" |
| 3 p_2^1 = |
<math>\begin{array}{lll}
| p_(p_1)^1 p_p @ @ ((())())
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
| ^
\end{array}</math>
| \
| width="14%" | <math>\text{p}_\text{p}\!</math>
| width="19%" | [[Image:Riff 3 Big.jpg|40px]]
| width="19%" | [[Image:Rote 3 Big.jpg|40px]]
| width="19%" | <math>(((~))(~))</math>
|-
| <math>4\!</math>
|
|
| o---o
<math>\begin{array}{lll}
| o |
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
| ^ o---o
\end{array}</math>
| 4 p_1^2 = / |
| <math>\text{p}^\text{p}\!</math>
| p_1^p_1 p^p @ @ (((())))
| [[Image:Riff 4 Big.jpg|40px]]
| [[Image:Rote 4 Big.jpg|65px]]
| <math>((((~))))</math>
|}
|-
|
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>5\!</math>
| width="19%" |
<math>\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}</math>
| width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="19%" | [[Image:Riff 5 Big.jpg|65px]]
| width="19%" | [[Image:Rote 5 Big.jpg|40px]]
| width="19%" | <math>((((~))(~))(~))</math>
|-
| <math>6\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
| o---o
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
| |
\end{array}</math>
| 5 p_3 = o---o
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| [[Image:Riff 6 Big.jpg|65px]]
| p_(p_(p_1)) p_(p_p) @ @ (((())())())
| [[Image:Rote 6 Big.jpg|80px]]
| ^
| <math>((~))(((~))(~))</math>
| \
|-
| <math>7\!</math>
|
|
<math>\begin{array}{lll}
| /
\text{p}_4^1
| o-o o-o
& = & \text{p}_{\text{p}_1^2}^1
| 6 p_1 p_2 = \ /
\\[10pt]
| p_1 p_(p_1) p p_p @ @ @ (())((())())
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
| ^
\end{array}</math>
| \
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 7 Big.jpg|65px]]
| [[Image:Rote 7 Big.jpg|65px]]
| <math>(((((~))))(~))</math>
|-
| <math>8\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_1^3
| o---o
& = & \text{p}_1^{\text{p}_2^1}
| |
\\[10pt]
| 7 p_4 = o---o
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}</math>
| p_(p_1^p_1) p_(p^p) @ o @ ((((())))())
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| ^ ^
| [[Image:Riff 8 Big.jpg|65px]]
| \ /
| [[Image:Rote 8 Big.jpg|65px]]
| <math>(((((~))(~))))</math>
|-
| <math>9\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| p_1^p_2 = / \ |
| [[Image:Riff 9 Big.jpg|40px]]
| p_1^p_(p_1) p^p_p @ o @ ((((())())))
| [[Image:Rote 9 Big.jpg|80px]]
| <math>(((~))(((~))))</math>
|-
| <math>16\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
| p_(p_1)^2 = / |
\\[10pt]
| p_(p_1)^(p_1) p_p^p @ @ ((())((())))
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
| ^
\end{array}</math>
| \
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| o
| [[Image:Riff 16 Big.jpg|65px]]
| [[Image:Rote 16 Big.jpg|90px]]
| <math>((((((~))))))</math>
|}
|}
====Small Scale====
{| align="center" border="1" width="96%"
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
|- style="height:50px; background:#f0f0ff"
|
|
| o o---o
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
| ^ |
| width="10%" | <math>\text{Integer}\!</math>
| / o---o
| width="19%" | <math>\text{Factorization}\!</math>
| o |
| width="14%" | <math>\text{Notation}\!</math>
| 16 p_1^4 = ^ o---o
| width="19%" | <math>\text{Riff Digraph}\!</math>
| p_1^(p_1^2) = / |
| width="19%" | <math>\text{Rote Graph}\!</math>
| p_1^(p_1^p_1) p^(p^p) @ @ (((((())))))
| width="19%" | <math>\text{Traversal}\!</math>
|}
|-
|
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>1\!</math>
| width="19%" | <math>1\!</math>
| width="14%" |
| width="19%" |
| width="19%" | [[Image:Rote 1 Big.jpg|12px]]
| width="19%" |
|}
|-
| jj
| p<
| j / ji
| p< p< etc.
| i \ ij
| p<
| ii
|
|
| = 1 + x.x^0 (1 + x + 2x^2 + ...)
{| cellpadding="12" style="text-align:center; width:100%"
| . 1 + x.x^1 (1 + x + 2x^2 + ...)
| width="10%" | <math>2\!</math>
| . 1 + x.x^2 (1 + x + 2x^2 + ...)
| width="19%" | <math>\text{p}_1^1\!</math>
| . 1 + x.x^2 (1 + x + 2x^2 + ...)
| width="14%" | <math>\text{p}\!</math>
| . ...
| width="19%" | [[Image:Riff 2 Big.jpg|12px]]
| width="19%" | [[Image:Rote 2 Big.jpg|24px]]
| width="19%" | <math>((~))</math>
|}
|-
|
|
| = 1 + x + 2x^2 + ...
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>3\!</math>
| width="19%" |
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| width="14%" | <math>\text{p}_\text{p}\!</math>
| width="19%" | [[Image:Riff 3 Big.jpg|24px]]
| width="19%" | [[Image:Rote 3 Big.jpg|24px]]
| width="19%" | <math>(((~))(~))</math>
|-
| <math>4\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
-------------------------------------------------------
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| [[Image:Riff 4 Big.jpg|24px]]
| [[Image:Rote 4 Big.jpg|38px]]
| <math>((((~))))</math>
|}
|-
|
|
| p
{| cellpadding="12" style="text-align:center; width:100%"
| p< p_p p p
| width="10%" | <math>5\!</math>
| width="19%" |
| p< p< p< p< p< p<
<math>\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}</math>
| width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="19%" | [[Image:Riff 5 Big.jpg|38px]]
| width="19%" | [[Image:Rote 5 Big.jpg|24px]]
| width="19%" | <math>((((~))(~))(~))</math>
|-
| <math>6\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| [[Image:Riff 6 Big.jpg|38px]]
| [[Image:Rote 6 Big.jpg|48px]]
| <math>((~))(((~))(~))</math>
|-
| <math>7\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_4^1
| p_p< p_p< p< p_p<_p p_p_p_p p_p_p<
& = & \text{p}_{\text{p}_1^2}^1
| p p_p
\\[10pt]
|
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 7 Big.jpg|38px]]
| [[Image:Rote 7 Big.jpg|38px]]
| <math>(((((~))))(~))</math>
|-
| <math>8\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_1^3
| p< p< p< p< p< p< p p<
& = & \text{p}_1^{\text{p}_2^1}
| p p p_p p^p p p
\\[10pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| [[Image:Riff 8 Big.jpg|38px]]
| [[Image:Rote 8 Big.jpg|38px]]
| <math>(((((~))(~))))</math>
|-
| <math>9\!</math>
|
|
<math>\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| [[Image:Riff 9 Big.jpg|24px]]
| [[Image:Rote 9 Big.jpg|48px]]
| <math>(((~))(((~))))</math>
|-
| <math>16\!</math>
|
|
<math>\begin{array}{lll}
| p^p
\text{p}_1^4
|
& = & \text{p}_1^{\text{p}_1^2}
\\[10pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 16 Big.jpg|38px]]
| [[Image:Rote 16 Big.jpg|52px]]
| <math>((((((~))))))</math>
|}
|}
===Old ASCII Version===
<pre>
Illustration of initial terms of A061396
Jon Awbrey (jawbrey(AT)oakland.edu)
o--------------------------------------------------------------------------------
| integer factorization riff r.i.f.f. rote --> in parentheses
o----------------------------------------------------------------------
| k p's k nodes 2k+1 nodes
o--------------------------------------------------------------------------------
|
|
| p
| 1 1 blank blank @ blank
|
|
o--------------------------------------------------------------------------------
o----------------------------------------------------------------------
|
|
| p
| o---o
| p< p_p p p
| |
| p_p< p_p< p< p_p<_p p_p_p_p p_p_p<
| 2 p_1^1 p @ @ (())
| p p_p
|
|
o--------------------------------------------------------------------------------
o----------------------------------------------------------------------
|
|
| p
| o---o
| p< p_p p p p
| |
| p< p< p< p< p^p p_p p p<
| o---o
| p p p_p p^p p
| 3 p_2^1 = |
| p_(p_1)^1 p_p @ @ ((())())
| ^
| \
| o
|
|
| 3^4 3^3 5^2 7^2
| o---o
| 81 27 25 49 12 18
| o |
| ^ o---o
| 4 p_1^2 = / |
| p_1^p_1 p^p @ @ (((())))
|
|
o--------------------------------------------------------------------------------
o--------------------------------------------------------------------------------
|
|
| o---o
| o---o
| |
| |
| o---o
| |
| 5 p_3 = o---o
| p_(p_2) = |
| o---o
| p_(p_(p_1)) p_(p_p) @ @ (((())())())
| |
| o---o
| 3 p_2^1 = |
| p_(p_1)^1 p_p @ @ ((())())
| ^
| ^
| \
| \
| o
| o
| ^
| ^
| \
| \
| o
| o
| 7 p_4 = o---o
| 7 p_4 = o---o
| p_(p_1^2) = |
| p_(p_1^2) = |
| p_(p_1^p_1) p< @ o @ ((((())))())
| p_(p_1^p_1) p_(p^p) @ o @ ((((())))())
| p^p ^ ^
| ^ ^
| \ /
| \ /
| o
| o
| o |
| o |
| 8 p_1^3 = ^ ^ o---o
| 8 p_1^3 = ^ ^ o---o
| p_1^p_2 = p_p / \ |
| p_1^p_2 = / \ |
| p_1^p_(p_1) p< @ o @ ((((())())))
| p_1^p_(p_1) p^p_p @ o @ ((((())())))
|
|
| o-o o-o
| o-o o-o
| o | |
| o | |
| 9 p_2^2 = ^ o---o
| 9 p_2^2 = ^ o---o
| p_(p_1)^2 = p / |
| p_(p_1)^2 = / |
| p_(p_1)^(p_1) p< @ @ ((())((())))
| p_(p_1)^(p_1) p_p^p @ @ ((())((())))
| p ^
| ^
| \
| \
| o
| o
| / o---o
| / o---o
| o |
| o |
| 16 p_1^4 = p ^ o---o
| 16 p_1^4 = ^ o---o
| p_1^(p_1^2) = p< / |
| p_1^(p_1^2) = / |
| p_1^(p_1^p_1) p< @ @ (((((())))))
| p_1^(p_1^p_1) p^(p^p) @ @ (((((())))))
|
|
o--------------------------------------------------------------------------------
o--------------------------------------------------------------------------------
Further Comments:
Here are a couple more pages from my notes,
where it looks like I first arrived at the
generating function, and also carried out
some brute force enumerations of riffs.
I am going to experiment with a different way of
transcribing indices and powers into a plaintext.
| jj
| integer factorization riff r.i.f.f. rote --> in parentheses
| p<
| k p's k nodes 2k+1 nodes
| j / ji
| p< p< etc.
| i \ ij
| p<
| ii
-------------------------------------------------------
1978-11-06
Generating Function
| R(x) = 1 + x + 2x^2 + ...
|
|
| 1 1 blank blank @ blank
| = 1 + x.x^0 (1 + x + 2x^2 + ...)
| . 1 + x.x^1 (1 + x + 2x^2 + ...)
| . 1 + x.x^2 (1 + x + 2x^2 + ...)
| . 1 + x.x^2 (1 + x + 2x^2 + ...)
| . ...
|
|
| = 1 + x + 2x^2 + ...
|
|
| o---o
| Product over (i = 0 to infinity) of (1 + x.x^i.R(x))^R_i = R(x)
| 2 p_1^1 p @ @ (())
-------------------------------------------------------
1978-11-10
Brute force enumeration of R_n
| 4 p's
|
|
| p
| p< p_p p p
| p< p< p p_p p<_p p_p_p p_p<
| p< p< p< p< p< p<
|
|
|
|
| o---o
| p
| o |
| p< p_p p p
| ^ o---o
| p_p< p_p< p< p_p<_p p_p_p_p p_p_p<
| 4 p_1^2 = / |
| p p_p
| p_1^p_1 p^p @ @ (((())))
|
|
|
| p
| p< p_p p p p p
| p< p< p< p< p< p< p p<
| p p p_p p^p p p
|
|
|
|
| o-o
| p p_p_p p p<
| p^p
| p_1 p_(p_1) p p_p @ @ @ (())((())())
|
|
| o---o
| |
Altogether, 20 riffs of weight 4.
| o---o
| |
| o---------------------o---------------------o---------------------o
| 7 p_4 = o---o
| | 3 | 4 | 5 |
| p_(p_1^2) = |
| o---------------------o---------------------o---------------------|
| p_(p_1^p_1) p< @ o @ ((((())))())
| | // // 2 | 10, 3, 1, 6 | 36, 10, 2, 3, 2, 20 |
| p^p ^ ^
| o---------------------o---------------------o---------------------|
| \ /
| | | 0^1 4^1, | |
| o
| | | 1^1 3^1, | |
| | | 2^2, | |
| | | 4^1 0^1 | |
| o---------------------o---------------------o---------------------o
| | 6 | 20 | 73 |
| o---------------------o---------------------o---------------------o
|
|
-------------------------------------------------------
Here are the number values of the riffs on 4 nodes:
o----------------------------------------------------------------------
|
|
| o-o o-o
| p
| o | |
| p< p_p p p
| 9 p_2^2 = ^ o---o
| p< p< p p_p p<_p p_p_p p_p<
| p< p< p< p< p< p<
|
|
| o o---o
| 2^16 2^8 2^6 2^9 2^5 2^7
| 65536 256 64 512 32 128
o----------------------------------------------------------------------
| p_1^(p_1^p_1) p< @ @ (((((())))))
o--------------------------------------------------------------------------------
|
|
| p
| p
| p< p p_p p
| p< p_p p p
| p< p<_p p< p_p< p p_p p_p_p
| p_p< p_p< p< p_p<_p p_p_p_p p_p_p<
| p< p< p< p< p< p<
| p p_p
|
|
| 2^16 2^9 2^8 2^7 2^6 2^5
| p_16 p_8 p_6 p_9 p_5 p_7
| 65536 512 256 128 64 32
| 53 19 13 23 11 17
o----------------------------------------------------------------------
|
|
| p
| p< p_p p p p
| p< p< p< p< p^p p_p p p<
| p p p_p p^p p
|
|
| p
| 3^4 3^3 5^2 7^2
| p< p p_p p
| 81 27 25 49 12 18
o----------------------------------------------------------------------
|
|
| p_16 p_9 p_8 p_7 p_6 p_5
| p p_p_p p p<
| 53 23 19 17 13 11
| p^p
|
|
o--------------------------------------------------------------------------------
| 10 14
o----------------------------------------------------------------------
For ease of reference, I include the previous table
of smaller riffs and rotes, redone in the new style.
o--------------------------------------------------------------------------------
| integer factorization riff r.i.f.f. rote --> in parentheses
| k p's k nodes 2k+1 nodes
o--------------------------------------------------------------------------------
|
|
| p^p p_p p p
| 1 1 blank blank @ blank
|
|
o--------------------------------------------------------------------------------
o--------------------------------------------------------------------------------
|
|
| p
| o---o
| p p< p p< p^p p_p p p_p_p
| |
| p p^p
| 2 p_1^1 p @ @ (())
|
|
o--------------------------------------------------------------------------------
|
|
o================================================================================
| o---o
| |
| o---o
| 3 p_2^1 = |
| p_(p_1)^1 p_p @ @ ((())())
| ^
--o------------------------------------------------
| \
| o
|
| o---o
| o |
| ^ o---o
| 4 p_1^2 = / |
| p_1^p_1 p^p @ @ (((())))
|
o--------------------------------------------------------------------------------
|
|
| Value: 2; 3, 4; 5, 6 , 7, 8, 9, 16;
| o---o
| |
| o---o
| |
| 5 p_3 = o---o
| p_(p_2) = |
| p_(p_(p_1)) p_p_p @ @ (((())())())
| ^
| \
| o
| ^
| \
| o
|
|
| o-o
| /
| o-o o-o
| 6 p_1 p_2 = \ /
| p_1 p_(p_1) p p_p @ @ @ (())((())())
| ^
| \
| o
|
|
| o---o
| |
| o---o
| |
| 7 p_4 = o---o
| p_(p_1^2) = |
| p_(p_1^p_1) p< @ o @ ((((())))())
| p^p ^ ^
| \ /
| o
|
|
| o---o
| |
| o---o
| o |
| 8 p_1^3 = ^ ^ o---o
| p_1^p_2 = p_p / \ |
| p_1^p_(p_1) p< @ o @ ((((())())))
|
|
| o-o o-o
| o | |
| 9 p_2^2 = ^ o---o
| p_(p_1)^2 = p / |
| p_(p_1)^(p_1) p< @ @ ((())((())))
| p ^
| \
| o
|
|
| o o---o
| ^ |
| / o---o
| o |
| 16 p_1^4 = p ^ o---o
| p_1^(p_1^2) = p< / |
| p_1^(p_1^p_1) p< @ @ (((((())))))
|
|
o--------------------------------------------------------------------------------
(later)
Expanded version of first table:
o--------------------------------------------------------------------------------
| integer factorization riff r.i.f.f. rote --> in parentheses
| k p's k nodes 2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1 1 blank blank @ blank
|
o--------------------------------------------------------------------------------
|
| o---o
| |
| 2 p_1^1 p @ @ (())
|
o--------------------------------------------------------------------------------
|
| o---o
| |
| o---o
| 3 p_2^1 = |
| p_(p_1)^1 p_p @ @ ((())())
| ^
| \
| o
|
| o---o
| o |
| ^ o---o
| 4 p_1^2 = / |
| p_1^p_1 p^p @ @ (((())))
|
o--------------------------------------------------------------------------------
| valign="bottom" |
|
| o---o
| |
| o---o
| |
| 5 p_3 = o---o
| p_(p_2) = |
| p_(p_(p_1)) p_p_p @ @ (((())())())
| ^
| valign="bottom" |
| \
| o
| ^
|-
| \
| valign="bottom" |
| o
|
| o-o
| valign="bottom" |
| /
| o-o o-o
| 6 p_1 p_2 = \ /
| p_1 p_(p_1) p p_p @ @ @ (())((())())
| ^
| \
| o
|
| o---o
| valign="bottom" |
| |
| o---o
| |
| 7 p_4 = o---o
| valign="bottom" |
| p_(p_1^2) = |
| p_(p_1^p_1) p< @ o @ ((((())))())
| p^p ^ ^
| valign="bottom" |
| \ /
| o
|
| valign="bottom" |
| o---o
| |
| o---o
| valign="bottom" |
| o |
| 8 p_1^3 = ^ ^ o---o
| p_1^p_2 = p_p / \ |
| valign="bottom" |
| p_1^p_(p_1) p< @ o @ ((((())())))
|
| o-o o-o
| o | |
| valign="bottom" |
| 9 p_2^2 = ^ o---o
<p>[[Image:Rote 16 Big.jpg|90px]]</p><br>
| p_(p_1)^2 = p / |
| p_(p_1)^(p_1) p< @ @ ((())((())))
| valign="bottom" |
| p ^
| \
| o
| valign="bottom" |
|
| o o---o
| ^ |
| valign="bottom" |
| / o---o
| o |
| 16 p_1^4 = p ^ o---o
| p_1^(p_1^2) = p< / |
| p_1^(p_1^p_1) p< @ @ (((((())))))
|
o--------------------------------------------------------------------------------
o================================================================================
|
| p
| p< p p_p p
| p< p<_p p< p_p< p p_p p_p_p
| p< p< p< p< p< p<
|
| 2^16 2^9 2^8 2^7 2^6 2^5
| 65536 512 256 128 64 32
|
<p><math>\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}</math></p>
o--------------------------------------------------------------------------------
|
<p>[[Image:Rote 25 Big.jpg|80px]]</p><br>
| p
<p><math>\begin{array}{l} 3\!:\!2 \\ 25 \end{array}</math></p>
| p< p p_p p
|-
| p_p< p_p<_p p_p< p_p_p< p< p_p_p_p
| valign="bottom" |
| p p_p
|
| p_16 p_9 p_8 p_7 p_6 p_5
| 53 23 19 17 13 11
|
o--------------------------------------------------------------------------------
|
| p^p p_p p p
<p><math>\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}</math></p>
| p< p< p< p<
| valign="bottom" |
| p p p^p p_p
|
| 3^4 3^3 7^2 5^2
| 81 27 49 25
|
o--------------------------------------------------------------------------------
|
| p
| p p< p p< p^p p_p p p_p_p
<p><math>\begin{array}{l} 11\!:\!1 \\ 31 \end{array}</math></p>
| p p^p
| valign="bottom" |
|
| 18 14 12 10
|
o================================================================================
Triangle in which k-th row lists natural number
values for the collection of riffs with k nodes.
k | natural numbers n such that |riff(n)| = k
| valign="bottom" |
--o------------------------------------------------
0 | 1;
1 | 2;
2 | 3, 4;
3 | 5, 6, 7, 8, 9, 16;
4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27,
| 32, 49, 53, 64, 81, 128, 256, 512, 65536;
The natural number values for the riffs with
at most 3 pts are as follows (@'s are roots):
| o o o o
| | ^ | ^
| v | v |
| o o o o o o o o o
| | ^ | | | ^ | ^ ^
| v | v v v | v/ |
| Riff: @; @, @; @, @ @, @, @, @, @;
|
|-
| Value: 2; 3, 4; 5, 6 , 7, 8, 9, 16;
| valign="bottom" |
---------------------------------------------------
1, 2, 3, 4, 5, 6, 7, 8, 9, 16,
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536,
---------------------------------------------------
| valign="bottom" |
1; 2; 3, 4; 5, 6, 7, 8, 9, 16;
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536;
---------------------------------------------------
</pre>
==A062504==
* [http://oeis.org/wiki/A062504 A062504]
===TeX Array===
{| align="center"
|
<math>\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}</math>
|}
===JPEG===
{| align="center" border="1" width="90%"
|+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math>
|- style="height:50px; background:#f0f0ff"
|
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
| width="10%" | <math>\text{Integer}\!</math>
| width="25%" | <math>\text{Factorization}\!</math>
| width="15%" | <math>\text{Notation}\!</math>
| width="25%" | <math>\text{Riff Digraph}\!</math>
| 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]]
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>2\!</math>
| width="25%" | <math>\text{p}_1^1\!</math>
| width="15%" | <math>\text{p}\!</math>
| 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>
| width="15%" | <math>\text{p}_\text{p}\!</math>
| width="25%" | [[Image:Riff 3 Big.jpg|40px]]
| width="25%" | [[Image:Rote 3 Big.jpg|40px]]
|-
| <math>4\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| [[Image:Riff 4 Big.jpg|40px]]
| [[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}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
| width="15%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="25%" | [[Image:Riff 5 Big.jpg|65px]]
| width="25%" | [[Image:Rote 5 Big.jpg|40px]]
|-
| <math>6\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| [[Image:Riff 6 Big.jpg|65px]]
| [[Image:Rote 6 Big.jpg|80px]]
|-
| <math>7\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 7 Big.jpg|65px]]
| [[Image:Rote 7 Big.jpg|65px]]
|-
| <math>8\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| [[Image:Riff 8 Big.jpg|65px]]
| [[Image:Rote 8 Big.jpg|65px]]
|-
| <math>9\!</math>
|
<math>\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| [[Image:Riff 9 Big.jpg|40px]]
| [[Image:Rote 9 Big.jpg|80px]]
|-
| <math>16\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 16 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}
\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}</math>
| width="15%" | <math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math>
| width="25%" | [[Image:Riff 10 Big.jpg|90px]]
| width="25%" | [[Image:Rote 10 Big.jpg|80px]]
|-
| <math>11\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math>
| [[Image:Riff 11 Big.jpg|90px]]
| [[Image:Rote 11 Big.jpg|40px]]
|-
| <math>12\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}} \text{p}_{\text{p}}\!</math>
| [[Image:Riff 12 Big.jpg|65px]]
| [[Image:Rote 12 Big.jpg|105px]]
|-
| <math>13\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math>
| [[Image:Riff 13 Big.jpg|65px]]
| [[Image:Rote 13 Big.jpg|80px]]
|-
| <math>14\!</math>
|
<math>\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}</math>
| <math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 14 Big.jpg|90px]]
| [[Image:Rote 14 Big.jpg|105px]]
|-
| <math>17\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 17 Big.jpg|90px]]
| [[Image:Rote 17 Big.jpg|65px]]
|-
| <math>18\!</math>
|
<math>\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}</math>
| <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math>
| [[Image:Riff 18 Big.jpg|65px]]
| [[Image:Rote 18 Big.jpg|120px]]
|-
| <math>19\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math>
| [[Image:Riff 19 Big.jpg|90px]]
| [[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]]
|-
| <math>25\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math>
| [[Image:Riff 25 Big.jpg|65px]]
| [[Image:Rote 25 Big.jpg|80px]]
|-
| <math>27\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math>
| [[Image:Riff 27 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}
\\[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}</math>
| <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math>
| [[Image:Riff 32 Big.jpg|90px]]
| [[Image:Rote 32 Big.jpg|65px]]
|-
| <math>49\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math>
| [[Image:Riff 49 Big.jpg|65px]]
| [[Image:Rote 49 Big.jpg|80px]]
|-
| <math>53\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 53 Big.jpg|90px]]
| [[Image:Rote 53 Big.jpg|90px]]
|-
| <math>64\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
| [[Image:Riff 64 Big.jpg|65px]]
| [[Image:Rote 64 Big.jpg|105px]]
|-
| <math>81\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 81 Big.jpg|65px]]
| [[Image:Rote 81 Big.jpg|105px]]
|-
| <math>128\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 128 Big.jpg|90px]]
| [[Image:Rote 128 Big.jpg|90px]]
|-
| <math>256\!</math>
|
<math>\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}</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}{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}</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>
| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 65536 Big.jpg|90px]]
| [[Image:Rote 65536 Big.jpg|115px]]
|}
|}
===ASCII===
<pre>
Example
* k | natural numbers n such that |riff(n)| = k
* 0 | 1;
* 1 | 2;
* 2 | 3, 4;
* 3 | 5, 6, 7, 8, 9, 16;
* 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536;
* The natural number values for the riffs with at most 3 pts are as follows (x = root):
* .................o.......o..o.......o
* .................|.......^..|.......^
* .................v.......|..v.......|
* ...........o..o..o....o..o..o..o.o..o
* ...........|..^..|....|..|..^..|.^..^
* ...........v..|..v....v..v..|..v/...|
* Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
* Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;
</pre>
==A062537==
* [http://oeis.org/wiki/A062537 A062537]
===Wiki + TeX + JPEG===
{| align="center" border="1" cellpadding="10"
|+ style="height:25px" | <math>a(n) = \text{Number of Nodes in the Riff of}~ n</math>
| valign="bottom" |
<p> </p><br>
<p><math>1\!</math></p><br>
<p><math>a(1) ~=~ 0</math></p>
| valign="bottom" |
<p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
<p><math>\text{p}\!</math></p><br>
<p><math>a(2) ~=~ 1</math></p>
| valign="bottom" |
<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
<p><math>\text{p}_\text{p}\!</math></p><br>
<p><math>a(3) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Riff 4 Big.jpg|40px]]</p><br>
<p><math>\text{p}^\text{p}\!</math></p><br>
<p><math>a(4) ~=~ 2</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>a(5) ~=~ 3</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>a(6) ~=~ 3</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>a(7) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Riff 8 Big.jpg|65px]]</p><br>
<p><math>\text{p}^{\text{p}_{\text{p}}}\!</math></p><br>
<p><math>a(8) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Riff 9 Big.jpg|40px]]</p><br>
<p><math>\text{p}_\text{p}^\text{p}\!</math></p><br>
<p><math>a(9) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Riff 10 Big.jpg|90px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
<p><math>a(10) ~=~ 4</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 11 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!</math></p><br>
<p><math>a(11) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 12 Big.jpg|65px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br>
<p><math>a(12) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 13 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math></p><br>
<p><math>a(13) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 14 Big.jpg|90px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math></p><br>
<p><math>a(14) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 15 Big.jpg|90px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
<p><math>a(15) ~=~ 5</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 16 Big.jpg|65px]]</p><br>
<p><math>\text{p}^{\text{p}^{\text{p}}}\!</math></p><br>
<p><math>a(16) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Riff 17 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math></p><br>
<p><math>a(17) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 18 Big.jpg|65px]]</p><br>
<p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
<p><math>a(18) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 19 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math></p><br>
<p><math>a(19) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 20 Big.jpg|90px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
<p><math>a(20) ~=~ 5</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 21 Big.jpg|90px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(21) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 22 Big.jpg|115px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(22) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 23 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
<p><math>a(23) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 24 Big.jpg|115px]]</p><br>
<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br>
<p><math>a(24) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 25 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
<p><math>a(25) ~=~ 4</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 26 Big.jpg|90px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(26) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 27 Big.jpg|65px]]</p><br>
<p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(27) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 28 Big.jpg|90px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(28) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 29 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(29) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 30 Big.jpg|115px]]</p><br>
<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(30) ~=~ 6</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 31 Big.jpg|115px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br>
<p><math>a(31) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 32 Big.jpg|90px]]</p><br>
<p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(32) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 33 Big.jpg|115px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(33) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 34 Big.jpg|115px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(34) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 35 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(35) ~=~ 6</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 36 Big.jpg|65px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
<p><math>a(36) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 37 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(37) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 38 Big.jpg|115px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(38) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 39 Big.jpg|115px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(39) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 40 Big.jpg|135px]]</p><br>
<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(40) ~=~ 6</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 41 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br>
<p><math>a(41) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 42 Big.jpg|115px]]</p><br>
<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(42) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 43 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(43) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 44 Big.jpg|115px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(44) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 45 Big.jpg|90px]]</p><br>
<p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(45) ~=~ 6</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 46 Big.jpg|90px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
<p><math>a(46) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 47 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(47) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 48 Big.jpg|65px]]</p><br>
<p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br>
<p><math>a(48) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 49 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br>
<p><math>a(49) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 50 Big.jpg|90px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
<p><math>a(50) ~=~ 5</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 51 Big.jpg|115px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(51) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 52 Big.jpg|90px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(52) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 53 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(53) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Riff 54 Big.jpg|90px]]</p><br>
<p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(54) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 55 Big.jpg|115px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(55) ~=~ 7</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 56 Big.jpg|135px]]</p><br>
<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(56) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 57 Big.jpg|115px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(57) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 58 Big.jpg|115px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(58) ~=~ 6</math></p>
| valign="bottom" |
<p>[[Image:Riff 59 Big.jpg|115px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br>
<p><math>a(59) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 60 Big.jpg|115px]]</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) ~=~ 7</math></p>
|}
==A062860==
* [http://oeis.org/wiki/A062860 A062860]
===Wiki + TeX + JPEG===
{| align="center" border="1" cellpadding="10"
|+ style="height:25px" | <math>a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}</math>
| valign="bottom" |
<p> </p><br>
<p><math>1\!</math></p><br>
<p><math>a(0) ~=~ 1</math></p>
| valign="bottom" |
<p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
<p><math>\text{p}\!</math></p><br>
<p><math>a(1) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
<p><math>\text{p}_\text{p}\!</math></p><br>
<p><math>a(2) ~=~ 3</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>a(3) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Riff 10 Big.jpg|90px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
<p><math>a(4) ~=~ 10</math></p>
|-
| valign="bottom" |
<p>[[Image:Riff 15 Big.jpg|90px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
<p><math>a(5) ~=~ 15</math></p>
| valign="bottom" |
<p>[[Image:Riff 30 Big.jpg|115px]]</p><br>
<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(6) ~=~ 30</math></p>
| valign="bottom" |
<p>[[Image:Riff 55 Big.jpg|115px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(7) ~=~ 55</math></p>
| valign="bottom" |
<p>[[Image:Riff 105 Big.jpg|115px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(8) ~=~ 105</math></p>
| valign="bottom" |
<p>[[Image:Riff 165 Big.jpg|135px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(9) ~=~ 165</math></p>
|}
==A106177==
* [http://oeis.org/wiki/A106177 A106177]
===Primal Codes of Finite Partial Functions on Positive Integers===
{| align="center"
|
<math>\begin{array}{rcl}
1 & = & \varnothing \\
2 & = & 1\!:\!1 \\
3 & = & 2\!:\!1 \\
4 & = & 1\!:\!2 \\
5 & = & 3\!:\!1 \\
6 & = & 1\!:\!1 ~~ 2\!:\!1 \\
7 & = & 4\!:\!1 \\
8 & = & 1\!:\!3 \\
9 & = & 2\!:\!2 \\
10 & = & 1\!:\!1 ~~ 3\!:\!1 \\
11 & = & 5\!:\!1 \\
12 & = & 1\!:\!2 ~~ 2\!:\!1 \\
13 & = & 6\!:\!1 \\
14 & = & 1\!:\!1 ~~ 4\!:\!1 \\
15 & = & 2\!:\!1 ~~ 3\!:\!1 \\
16 & = & 1\!:\!4 \\
17 & = & 7\!:\!1 \\
18 & = & 1\!:\!1 ~~ 2\!:\!2 \\
19 & = & 8\!:\!1 \\
20 & = & 1\!:\!2 ~~ 3\!:\!1
\end{array}</math>
|}
===Wiki Table===
{| align="center" style="font-weight:bold; text-align:center"
| || || || || || || || ||
| <font color="red">1</font>
|
| <font color="red">1</font>
|-
| || || || || || || ||
| <font color="red">2</font>
| || 1 ||
| <font color="red">2</font>
|-
| || || || || || ||
| <font color="red">3</font>
| || 1 || || 1 ||
| <font color="red">3</font>
|-
| || || || || ||
| <font color="red">4</font>
| || 1 || || 2 || || 1 ||
| <font color="red">4</font>
|-
| || || || ||
| <font color="red">5</font>
| || 1 || || 3 || || 1 || || 1 ||
| <font color="red">5</font>
|-
| || || ||
| <font color="red">6</font>
| || 1 || || 1 || || 1 || || 4 || || 1 ||
| <font color="red">6</font>
|-
| || ||
| <font color="red">7</font>
| || 1 || || 5 || || 2 || || 9 || || 1 || || 1 ||
| <font color="red">7</font>
|-
| ||
| <font color="red">8</font>
| || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">8</font>
|-
|
| <font color="red">9</font>
| || 1 || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || 1 ||
| <font color="red">9</font>
|-
| width="12pt" | <font color="red">10</font>
| width="12pt" |
| width="12pt" | 1
| width="12pt" |
| width="12pt" | 1
| width="12pt" |
| width="12pt" | 1
| width="12pt" |
| width="12pt" | 36
| width="12pt" |
| width="12pt" | 1
| width="12pt" |
| width="12pt" | 2
| width="12pt" |
| width="12pt" | 1
| width="12pt" |
| width="12pt" | 8
| width="12pt" |
| width="12pt" | 1
| width="12pt" |
| width="12pt" | <font color="red">10</font>
|}
===Wiki + TeX===
====Smallmatrix====
{| align="center"
|
<math>\begin{smallmatrix}
& & & & & & & & & {\color{red}1} & & {\color{red}1}
\\
& & & & & & & & {\color{red}2} & & 1 & & {\color{red}2}
\\
& & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3}
\\
& & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4}
\\
& & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5}
\\
& & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6}
\\
& & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7}
\\
& & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8}
\\
& {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9}
\\
{\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10}
\end{smallmatrix}</math>
|}
====Array====
{| align="center"
|
<math>\begin{array}{*{21}{c}}
& & & & & & & & & {\color{red}1} & & {\color{red}1}
\\
& & & & & & & & {\color{red}2} & & 1 & & {\color{red}2}
\\
& & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3}
\\
& & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4}
\\
& & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5}
\\
& & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6}
\\
& & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7}
\\
& & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8}
\\
& {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9}
\\
{\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10}
\end{array}</math>
|}
====Matrix====
{| align="center"
|
<math>\begin{matrix}
n \circ m
\\
1 ~/~\backslash~ 1
\\
2 ~/~ 1 ~\backslash~ 2
\\
3 ~/~ 1 \cdot 1 ~\backslash~ 3
\\
4 ~/~ 1 \cdot 2 \cdot 1 ~\backslash~ 4
\\
5 ~/~ 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 5
\\
6 ~/~ 1 \cdot 1 \cdot 1 \cdot 4 \cdot 1 ~\backslash~ 6
\\
7 ~/~ 1 \cdot 5 \cdot 2 \cdot 9 \cdot 1 \cdot 1 ~\backslash~ 7
\\
8 ~/~ 1 \cdot 6 \cdot 1 \cdot 1 \cdot 1 \cdot 2 \cdot 1 ~\backslash~ 8
\\
9 ~/~ 1 \cdot 7 \cdot 1 \cdot 25\cdot 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 9
\\
10 ~/~ 1 \cdot 1 \cdot 1 \cdot 36\cdot 1 \cdot 2 \cdot 1 \cdot 8 \cdot 1 ~\backslash~ 10
\end{matrix}</math>
|}
===ASCII===
<pre>
Example
* n o m
* \ /
* 1 . 1
* \ / \ /
* 2 . 1 . 2
* \ / \ / \ /
* 3 . 1 . 1 . 3
* \ / \ / \ / \ /
* 4 . 1 . 2 . 1 . 4
* \ / \ / \ / \ / \ /
* 5 . 1 . 3 . 1 . 1 . 5
* \ / \ / \ / \ / \ / \ /
* 6 . 1 . 1 . 1 . 4 . 1 . 6
* \ / \ / \ / \ / \ / \ / \ /
* 7 . 1 . 5 . 2 . 9 . 1 . 1 . 7
* \ / \ / \ / \ / \ / \ / \ / \ /
* 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8
* \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10
*
* Primal codes of finite partial functions on positive integers:
* 1 = { }
* 2 = 1:1
* 3 = 2:1
* 4 = 1:2
* 5 = 3:1
* 6 = 1:1 2:1
* 7 = 4:1
* 8 = 1:3
* 9 = 2:2
* 10 = 1:1 3:1
* 11 = 5:1
* 12 = 1:2 2:1
* 13 = 6:1
* 14 = 1:1 4:1
* 15 = 2:1 3:1
* 16 = 1:4
* 17 = 7:1
* 18 = 1:1 2:2
* 19 = 8:1
* 20 = 1:2 3:1
</pre>
==A106178==
* [http://oeis.org/wiki/A106178 A106178]
===Wiki Table===
{| align="center" style="font-weight:bold; text-align:center; width:90%"
| || || || || || || || || || || || || || ||
| <font color="red">1</font>
|
| <font color="red">1</font>
|-
| || || || || || || || || || || || || ||
| <font color="red">2</font>
| || · ||
| <font color="red">2</font>
|-
| || || || || || || || || || || || ||
| <font color="red">3</font>
| || · || || · ||
| <font color="red">3</font>
|-
| || || || || || || || || || || ||
| <font color="red">4</font>
| || · || || 2 || || · ||
| <font color="red">4</font>
|-
| || || || || || || || || || ||
| <font color="red">5</font>
| || · || || 3 || || 1 || || · ||
| <font color="red">5</font>
|-
| || || || || || || || || ||
| <font color="red">6</font>
| || · || || 1 || || 1 || || 4 || || · ||
| <font color="red">6</font>
|-
| || || || || || || || ||
| <font color="red">7</font>
| || · || || 5 || || 2 || || 9 || || 1 || || · ||
| <font color="red">7</font>
|-
| || || || || || || ||
| <font color="red">8</font>
| || · || || 6 || || 1 || || 1 || || 1 || || 2 || || · ||
| <font color="red">8</font>
|-
| || || || || || ||
| <font color="red">9</font>
| || · || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || · ||
| <font color="red">9</font>
|-
| || || || || ||
| <font color="red">10</font>
| || · || || 1 || || 1 || || 36|| || 1 || || 2 || || 1 || || 8 || || · ||
| <font color="red">10</font>
|-
| || || || ||
| <font color="red">11</font>
| || · || || 1 || || 1 || || 49 || || 1 || || 5 || || 1 || || 27 || || 1 || || · ||
| <font color="red">11</font>
|-
| || || ||
| <font color="red">12</font>
| || · || || 10 || || 3 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || · ||
| <font color="red">12</font>
|-
| || ||
| <font color="red">13</font>
| || · || || 11 || || 1 || || 1 || || 2 || || 7 || || 1 || || 125 || || 4 || || 3 || || 1 || || · ||
| <font color="red">13</font>
|-
| ||
| <font color="red">14</font>
| || · || || 3 || || 1 || || 100 || || 1 || || 1 || || 1 || || 216 || || 1 || || 1 || || 1 || || 4 || || · ||
| <font color="red">14</font>
|-
|
| <font color="red">15</font>
| || · || || 13 || || 2 || || 121 || || 1 || || 3 || || 1 || || 343 || || 1 || || 5 || || 1 || || 9 || || 1 || || · ||
| <font color="red">15</font>
|-
| width="3%" | <font color="red">16</font>
| width="3%" |
| width="3%" | ·
| width="3%" |
| width="3%" | 14
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 9
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 10
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 6
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 2
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 2
| width="3%" |
| width="3%" | ·
| width="3%" |
| width="3%" | <font color="red">16</font>
|}
===TeX Smallmatrix===
{| align="center"
|
<math>\begin{smallmatrix}
&&&&&&&&&&&&&&& {\color{red}1} && {\color{red}1}
\\
&&&&&&&&&&&&&& {\color{red}2} && \cdot & & {\color{red}2}
\\
&&&&&&&&&&&&& {\color{red}3} && \cdot && \cdot && {\color{red}3}
\\
&&&&&&&&&&&& {\color{red}4} && \cdot && 2 && \cdot && {\color{red}4}
\\
&&&&&&&&&&& {\color{red}5} && \cdot && 3 && 1 && \cdot && {\color{red}5}
\\
&&&&&&&&&& {\color{red}6} && \cdot && 1 && 1 && 4 && \cdot && {\color{red}6}
\\
&&&&&&&&& {\color{red}7} && \cdot && 5 && 2 && 9 && 1 && \cdot && {\color{red}7}
\\
&&&&&&&& {\color{red}8} && \cdot && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}8}
\\
&&&&&&& {\color{red}9} && \cdot && 7 && 1 && 25 && 1 && 3 && 1 && \cdot && {\color{red}9}
\\
&&&&&& {\color{red}10} && \cdot && 1 && 1 && 36 && 1 && 2 && 1 && 8 && \cdot && {\color{red}10}
\\
&&&&& {\color{red}11} && \cdot && 1 && 1 && 49 && 1 && 5 && 1 && 27 && 1 && \cdot && {\color{red}11}
\\
&&&& {\color{red}12} && \cdot && 10 && 3 && 1 && 1 && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}12}
\\
&&& {\color{red}13} && \cdot && 11 && 1 && 1 && 2 && 7 && 1 && 125 && 4 && 3 && 1 && \cdot && {\color{red}13}
\\
&& {\color{red}14} && \cdot && 3 && 1 && 100 && 1 && 1 && 1 && 216 && 1 && 1 && 1 && 4 && \cdot && {\color{red}14}
\\
& {\color{red}15} && \cdot && 13 && 2 && 121 && 1 && 3 && 1 && 343 && 1 && 5 && 1 && 9 && 1 && \cdot && {\color{red}15}
\\
{\color{red}16} && \cdot && 14 && 1 && 9 && 1 && 10 && 1 && 1 && 1 && 6 && 1 && 2 && 1 && 2 && \cdot && {\color{red}16}
\end{smallmatrix}</math>
|}
===ASCII===
<pre>
Example
* n o m
* \ /
* 1 . 1
* \ / \ /
* 2 . . 2
* \ / \ / \ /
* 3 . . . 3
* \ / \ / \ / \ /
* 4 . . 2 . . 4
* \ / \ / \ / \ / \ /
* 5 . . 3 . 1 . . 5
* \ / \ / \ / \ / \ / \ /
* 6 . . 1 . 1 . 4 . . 6
* \ / \ / \ / \ / \ / \ / \ /
* 7 . . 5 . 2 . 9 . 1 . . 7
* \ / \ / \ / \ / \ / \ / \ / \ /
* 8 . . 6 . 1 . 1 . 1 . 2 . . 8
* \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 9 . . 7 . 1 . 25. 1 . 3 . 1 . . 9
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 10 . . 1 . 1 . 36. 1 . 2 . 1 . 8 . . 10
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 11 . . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 . . 11
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 12 . . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . . 12
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 13 . . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 . . 13
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 14 . . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 . . 14
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 15 . . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 . . 15
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 16 . . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 . . 16
</pre>
==A108352==
* [http://oeis.org/wiki/A108352 A108352]
===Links===
* Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002846.html Primal Code Characteristic, n = 1 to 1000]
* Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002847.html Primal Code Characteristic, n = 1001 to 2000]
* Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002853.html Primal Code Characteristic, n = 2001 to 3000]
===TeX Array===
{| align="center"
|
<math>\begin{array}{*{10}{l}}
a(1)
& = & 1
& \text{because} & (\circ~ 1)^1
& = & (\circ~ \varnothing)^1
& = & 1.
\\
a(2)
& = & 0
& \text{because} & (\circ~ 2)^k
& = & (\circ~ 1\!:\!1)^k
& = & 2,
& \text{for all}~ k > 0.
\\
a(3)
& = & 2
& \text{because} & (\circ~ 3)^2
& = & (\circ~ 2\!:\!1)^2
& = & 1.
\\
a(4)
& = & 2
& \text{because} & (\circ~ 4 )^2
& = & (\circ~ 1\!:\!2)^2
& = &1.
\\
a(5)
& = & 2
& \text{because} & (\circ~ 5)^2
& = & (\circ~ 3\!:\!1)^2
& = & 1.
\\
a(6)
& = & 0
& \text{because} & (\circ~ 6)^k
& = & (\circ~ 1\!:\!1 ~~ 2\!:\!1)^k
& = & 6,
& \text{for all}~ k > 0.
\\
a(7)
& = & 2
& \text{because} & (\circ~ 7)^2
& = & (\circ~ 4\!:\!1)^1
& = & 1.
\\
a(8)
& = & 2
& \text{because} & (\circ~ 8)^2
& = & (\circ~ 1\!:\!3)^1
& = & 1.
\\
a(9)
& = & 0
& \text{because} & (\circ~ 9)^k
& = & (\circ~ 2\!:\!2)^k
& = & 9,
& \text{for all}~ k > 0.
\\
a(10)
& = & 0
& \text{because} & (\circ~ 10)^k
& = & (\circ~ 1\!:\!1 ~~ 3\!:\!1)^k
& = & 10,
& \text{for all}~ k > 0.
\end{array}</math>
|}
===ASCII===
<pre>
Example
* a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1.
* a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k.
* a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1.
* a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1.
* a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1.
* a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
* a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1.
* a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1.
* a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k.
* a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.
* Detail of calculation for compositional powers of 12:
* (12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
* (12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12
* Detail of calculation for compositional powers of 20:
* (20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
* (20 o)^3 = (3:2) o (1:2 3:1) = 1
</pre>
==A108371==
* [http://oeis.org/wiki/A108371 A108371]
===Wiki Table===
{| align="center" style="font-weight:bold; text-align:center; width:90%"
| || || || || || || || || || || || || || ||
| <font color="red">1</font>
|
| <font color="red">1</font>
|-
| || || || || || || || || || || || || ||
| <font color="red">2</font>
| || 1 ||
| <font color="red">2</font>
|-
| || || || || || || || || || || || ||
| <font color="red">3</font>
| || 2 || || 1 ||
| <font color="red">3</font>
|-
| || || || || || || || || || || ||
| <font color="red">4</font>
| || 3 || || 2 || || 1 ||
| <font color="red">4</font>
|-
| || || || || || || || || || ||
| <font color="red">5</font>
| || 4 || || 1 || || 2 || || 1 ||
| <font color="red">5</font>
|-
| || || || || || || || || ||
| <font color="red">6</font>
| || 5 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">6</font>
|-
| || || || || || || || ||
| <font color="red">7</font>
| || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">7</font>
|-
| || || || || || || ||
| <font color="red">8</font>
| || 7 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">8</font>
|-
| || || || || || ||
| <font color="red">9</font>
| || 8 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">9</font>
|-
| || || || || ||
| <font color="red">10</font>
| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">10</font>
|-
| || || || ||
| <font color="red">11</font>
| || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">11</font>
|-
| || || ||
| <font color="red">12</font>
| || 11|| || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">12</font>
|-
| || ||
| <font color="red">13</font>
| || 12|| || 1 || || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">13</font>
|-
| ||
| <font color="red">14</font>
| || 13|| || 18|| || 1 || || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">14</font>
|-
|
| <font color="red">15</font>
| || 14 || || 1 || || 12 || || 1 || || 10 || || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
| <font color="red">15</font>
|-
| width="3%" | <font color="red">16</font>
| width="3%" |
| width="3%" | 15
| width="3%" |
| width="3%" | 14
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 18
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 10
| width="3%" |
| width="3%" | 9
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 6
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | 2
| width="3%" |
| width="3%" | 1
| width="3%" |
| width="3%" | <font color="red">16</font>
|}
===ASCII===
<pre>
Example
* Table: T(n,k) = (n o)^k
* T(n,k)
* \ /
* 1 . 1
* \ / \ /
* 2 . 1 . 2
* \ / \ / \ /
* 3 . 2 . 1 . 3
* \ / \ / \ / \ /
* 4 . 3 . 2 . 1 . 4
* \ / \ / \ / \ / \ /
* 5 . 4 . 1 . 2 . 1 . 5
* \ / \ / \ / \ / \ / \ /
* 6 . 5 . 1 . 1 . 2 . 1 . 6
* \ / \ / \ / \ / \ / \ / \ /
* 7 . 6 . 1 . 1 . 1 . 2 . 1 . 7
* \ / \ / \ / \ / \ / \ / \ / \ /
* 8 . 7 . 6 . 1 . 1 . 1 . 2 . 1 . 8
* \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 9 . 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 9
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 10 . 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 10
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 11 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 11
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 12 . 11. 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 12
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 13 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 13
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 14 . 13. 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 14
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 15 . 14. 1 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 15
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 16 . 15. 14. 1 . 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 16
</pre>
==A109300==
* [http://oeis.org/wiki/A109300 A109300]
===JPEG===
{| align="center" border="1" cellpadding="10"
|
<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
|
<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
|
<p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
|
<p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p>
|
<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p>
|
<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p>
|
<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p>
|}
===ASCII===
<pre>
Example
* Table of Rotes and Primal Functions for Positive Integers of Rote Height 2
*
* o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | | | | | |
* o-o o-o o-o o-o o---o o-o o-o o-o o---o o-o o---o
* | | | | | | | | | | |
* O O O===O O O=====O O===O O=====O
*
* 2:1 1:2 1:1 2:1 2:2 1:2 2:1 1:1 2:2 1:2 2:2
*
* 3 4 6 9 12 18 36
*
</pre>
==A109301==
* [http://oeis.org/wiki/A109301 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>
===JPEG===
{| align="center" border="1" cellpadding="6"
| valign="bottom" |
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
<p><math>1\!</math></p><br>
<p><math>a(1) ~=~ 0</math></p>
| valign="bottom" |
<p>[[Image:Rote 2 Big.jpg|40px]]</p><br>
<p><math>\text{p}\!</math></p><br>
<p><math>a(2) ~=~ 1</math></p>
| valign="bottom" |
<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
<p><math>\text{p}_\text{p}\!</math></p><br>
<p><math>a(3) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
<p><math>\text{p}^\text{p}\!</math></p><br>
<p><math>a(4) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Rote 5 Big.jpg|40px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(5) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
<p><math>\text{p} \text{p}_\text{p}\!</math></p><br>
<p><math>a(6) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Rote 7 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(7) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 8 Big.jpg|65px]]</p><br>
<p><math>\text{p}^{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(8) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
<p><math>\text{p}_\text{p}^\text{p}\!</math></p><br>
<p><math>a(9) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Rote 10 Big.jpg|80px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(10) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 11 Big.jpg|40px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(11) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br>
<p><math>a(12) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Rote 13 Big.jpg|80px]]</p><br>
<p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(13) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 14 Big.jpg|105px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(14) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 15 Big.jpg|80px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(15) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 16 Big.jpg|90px]]</p><br>
<p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(16) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 17 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(17) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
<p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
<p><math>a(18) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Rote 19 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(19) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 20 Big.jpg|105px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(20) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 21 Big.jpg|105px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(21) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 22 Big.jpg|80px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(22) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 23 Big.jpg|80px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
<p><math>a(23) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 24 Big.jpg|105px]]</p><br>
<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br>
<p><math>a(24) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 25 Big.jpg|80px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
<p><math>a(25) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 26 Big.jpg|120px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(26) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 27 Big.jpg|80px]]</p><br>
<p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(27) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 28 Big.jpg|130px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(28) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 29 Big.jpg|80px]]</p><br>
<p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(29) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 30 Big.jpg|120px]]</p><br>
<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(30) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 31 Big.jpg|40px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br>
<p><math>a(31) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Rote 32 Big.jpg|65px]]</p><br>
<p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(32) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 33 Big.jpg|80px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(33) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 34 Big.jpg|105px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(34) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 35 Big.jpg|105px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(35) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
<p><math>a(36) ~=~ 2</math></p>
| valign="bottom" |
<p>[[Image:Rote 37 Big.jpg|105px]]</p><br>
<p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(37) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 38 Big.jpg|105px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(38) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 39 Big.jpg|120px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(39) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 40 Big.jpg|105px]]</p><br>
<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(40) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 41 Big.jpg|80px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br>
<p><math>a(41) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 42 Big.jpg|145px]]</p><br>
<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(42) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 43 Big.jpg|105px]]</p><br>
<p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(43) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 44 Big.jpg|105px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(44) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 45 Big.jpg|120px]]</p><br>
<p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(45) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 46 Big.jpg|120px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
<p><math>a(46) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 47 Big.jpg|80px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(47) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 48 Big.jpg|105px]]</p><br>
<p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br>
<p><math>a(48) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 49 Big.jpg|80px]]</p><br>
<p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br>
<p><math>a(49) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 50 Big.jpg|120px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
<p><math>a(50) ~=~ 3</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 51 Big.jpg|105px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(51) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 52 Big.jpg|145px]]</p><br>
<p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
<p><math>a(52) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 53 Big.jpg|90px]]</p><br>
<p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br>
<p><math>a(53) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 54 Big.jpg|120px]]</p><br>
<p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
<p><math>a(54) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 55 Big.jpg|80px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(55) ~=~ 4</math></p>
|-
| valign="bottom" |
<p>[[Image:Rote 56 Big.jpg|130px]]</p><br>
<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
<p><math>a(56) ~=~ 3</math></p>
| valign="bottom" |
<p>[[Image:Rote 57 Big.jpg|105px]]</p><br>
<p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(57) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 58 Big.jpg|120px]]</p><br>
<p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
<p><math>a(58) ~=~ 4</math></p>
| valign="bottom" |
<p>[[Image:Rote 59 Big.jpg|65px]]</p><br>
<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br>
<p><math>a(59) ~=~ 5</math></p>
| valign="bottom" |
<p>[[Image:Rote 60 Big.jpg|155px]]</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>
|}
===ASCII===
<pre>
Comment
* Table of Rotes and Primal Functions for Positive Integers from 1 to 40
*
*
* 36 37 38 39 40
* o-o
*
* |
* In these Figures, "extended lines of identity" like o===o
* o-o o-o o-o
* indicate identified nodes and capital O is the root node.
* | | |
* The rote height in gammas is found by finding the number
* o-o o-o o-o o-o
* of graphs of the following shape between the root and one
* | | | |
* of the highest nodes of the tree:
* O O O O O
* o--o
*
* |
* { } 1:1 2:1 1:2 3:1
* o
*
* A sequence like this, that can be regarded as a nonnegative integer
* 1 2 3 4 5
* measure on positive integers, may have as many as 3 other sequences
*
* associated with it. Given that the fiber of a function f at n is all
*
* the domain elements that map to n, we always have the fiber minimum
* o-o o-o o-o
* or minimum inverse function and may also have the fiber cardinality
* | | |
* and the fiber maximum or maximum inverse function. For A109301, the
* o-o o-o o-o o-o o-o o-o
* minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the
* | | | | | |
* first positive integer whose rote height is n, the fiber cardinality
* o-o o-o o-o o-o o---o o-o o-o
* is A109300, giving the number of positive integers of rote height n,
* | | | | | | |
* while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving
* O===O O O O O===O
* the last positive integer whose rote height is n, has the following
*
* initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36,
* 1:1 2:1 4:1 1:3 2:2 1:1 3:1
* while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
*
* (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
* 6 7 8 9 10
* 7.840858554516122655953405327738 x 10^371.
*
*
* o-o
* |
* o-o o-o o-o o-o
* | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | |
* o-o o-o o-o o===o-o o-o o-o o-o o-o
* | | | | | | | |
* O O=====O O O===O O===O
*
* 5:1 1:2 2:1 6:1 1:1 4:1 2:1 3:1
*
* 11 12 13 14 15
*
*
* o-o o-o
* | |
* o-o o-o o-o o-o
* | | | |
* o-o o-o o-o o-o o-o o-o o-o
* | | | | | | |
* o-o o-o o-o o---o o-o o-o o-o
* | | | | | | |
* O O O===O O O=====O
*
* 1:4 7:1 1:1 2:2 8:1 1:2 3:1
*
* 16 17 18 19 20
*
*
* o-o
* |
* o-o o-o o-o o-o o-o o-o
* | | | | | |
* o-o o-o o-o o---o o-o o-o o-o o-o
* | | | | | | | |
* o-o o-o o-o o-o o-o o-o o-o o---o
* | | | | | | | |
* O===O O===O O O=====O O
*
* 2:1 4:1 1:1 5:1 9:1 1:3 2:1 3:2
*
* 21 22 23 24 25
*
*
* o-o
* |
* o-o o-o o-o o-o o-o
* | | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | | | |
* o-o o===o-o o---o o-o o-o o===o-o o-o o-o o-o
* | | | | | | | | |
* O===O O O=====O O O===O===O
*
* 1:1 6:1 2:3 1:2 4:1 10:1 1:1 2:1 3:1
*
* 26 27 28 29 30
*
*
* o-o
* |
* o-o o-o o-o o-o
* | | | |
* o-o o-o o-o o-o o-o o-o
* | | | | | |
* o-o o-o o-o o-o o-o o-o o-o
* | | | | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | |
* O O O===O O===O O===O
*
* 11:1 1:5 2:1 5:1 1:1 7:1 3:1 4:1
*
* 31 32 33 34 35
*
*
* o-o
* |
* o-o o-o o-o o-o o-o o-o
* | | | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | | | | |
* o-o o---o o=====o-o o-o o-o o-o o===o-o o-o o-o
* | | | | | | | | |
* O=====O O O===O O===O O=====O
*
* 1:2 2:2 12:1 1:1 8:1 2:1 6:1 1:3 3:1
*
* 36 37 38 39 40
*
* In these Figures, "extended lines of identity" like o===o
* indicate identified nodes and capital O is the root node.
* The rote height in gammas is found by finding the number
* of graphs of the following shape between the root and one
* of the highest nodes of the tree:
* o--o
* |
* o
* A sequence like this, that can be regarded as a nonnegative integer
* measure on positive integers, may have as many as 3 other sequences
* associated with it. Given that the fiber of a function f at n is all
* the domain elements that map to n, we always have the fiber minimum
* or minimum inverse function and may also have the fiber cardinality
* and the fiber maximum or maximum inverse function. For A109301, the
* minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the
* first positive integer whose rote height is n, the fiber cardinality
* is A109300, giving the number of positive integers of rote height n,
* while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving
* the last positive integer whose rote height is n, has the following
* initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36,
* while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
* (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
* 7.840858554516122655953405327738 x 10^371.
Example
* Writing (prime(i))^j as i:j, we have:
* 802701 = 2:2 8638:1
* 8638 = 1:1 4:1 113:1
* 113 = 30:1
* 30 = 1:1 2:1 3:1
* 4 = 1:2
* 3 = 2:1
* 2 = 1:1
* 1 = { }
* So rote(802701) is the graph:
*
* o-o
* |
* o-o o-o
* | |
* o-o o-o o-o o-o
* | | | |
* o-o o===o===o-o
* | |
* o-o o-o o-o o-o o---------o
* | | | | |
* o---o o===o=====o---------o
* | |
* O=======O
*
* Therefore rhig(802701) = 6.
</pre>
</pre>
===JPEG===
===JPEG===
{| align="center" border="1" cellpadding="10"
{| align="center" border="1" cellpadding="10"
<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p>
<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p>
|}
|}
===ASCII===
===ASCII===
