Difference between revisions of "User:Jon Awbrey/SEQUENCES"

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(Undo revision 107256 by Jon Awbrey (Talk))
 
(76 intermediate revisions by the same user not shown)
Line 5: Line 5:
 
===Plain Wiki Table===
 
===Plain Wiki Table===
  
−
<br>
+
====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>
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| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
−
| [[Image:Rooted Node Big.jpg|20px]]
+
| [[Image:Rote 1 Big.jpg|20px]]
 
| &nbsp;
 
| &nbsp;
 
|-
 
|-
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| <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>
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\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>
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\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>
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\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>
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\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>
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\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>
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\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>
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\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>
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\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>
 
|}
 
|}
  
−
<br>
+
====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>
 +
| <math>\text{Factorization}\!</math>
 +
| <math>\text{Notation}\!</math>
 +
| <math>\text{Riff Digraph}\!</math>
 +
| <math>\text{Rote Graph}\!</math>
 +
| <math>\text{Traversal}\!</math>
 +
|- style="height:48px"
 +
| <math>1\!</math>
 +
| <math>1\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| [[Image:Rote 1 Big.jpg|12px]]
 +
| &nbsp;
 +
|-
 +
| <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]]
 +
| <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>
 +
|-
 +
| <math>5\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\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>
 +
| <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 5 Big.jpg|38px]]
 +
| [[Image:Rote 5 Big.jpg|24px]]
 +
| <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
 +
& = & \text{p}_{\text{p}_1^2}^1
 +
\\[6pt]
 +
& = & \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
 +
& = & \text{p}_1^{\text{p}_2^1}
 +
\\[6pt]
 +
& = & \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}
 +
\text{p}_1^4
 +
& = & \text{p}_1^{\text{p}_1^2}
 +
\\[6pt]
 +
& = & \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>
 +
|}
  
 
===Nested Wiki Table===
 
===Nested Wiki Table===
  
−
<br>
+
====Large Scale====
  
 
{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
−
|+ style="height:25px" | <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:50px; background:#f0f0ff"
 
|- style="height:50px; background:#f0f0ff"
 
|
 
|
Line 151: Line 270:
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
 
| width="19%" | &nbsp;
 
| width="19%" | &nbsp;
−
| width="19%" | [[Image:Rooted Node Big.jpg|20px]]
+
| width="19%" | [[Image:Rote 1 Big.jpg|20px]]
 
| width="19%" | &nbsp;
 
| width="19%" | &nbsp;
 
|}
 
|}
Line 160: Line 279:
 
| width="19%" | <math>\text{p}_1^1\!</math>
 
| width="19%" | <math>\text{p}_1^1\!</math>
 
| width="14%" | <math>\text{p}\!</math>
 
| width="14%" | <math>\text{p}\!</math>
−
| width="19%" | [[Image:Rooted Node Big.jpg|20px]]
+
| width="19%" | [[Image:Riff 2 Big.jpg|20px]]
 
| width="19%" | [[Image:Rote 2 Big.jpg|40px]]
 
| width="19%" | [[Image:Rote 2 Big.jpg|40px]]
 
| width="19%" | <math>((~))</math>
 
| width="19%" | <math>((~))</math>
Line 173: Line 292:
 
\end{array}</math>
 
\end{array}</math>
 
| width="14%" | <math>\text{p}_\text{p}\!</math>
 
| width="14%" | <math>\text{p}_\text{p}\!</math>
−
| width="19%" | <math>\cdots</math>
+
| width="19%" | [[Image:Riff 3 Big.jpg|40px]]
 
| width="19%" | [[Image:Rote 3 Big.jpg|40px]]
 
| width="19%" | [[Image:Rote 3 Big.jpg|40px]]
 
| width="19%" | <math>(((~))(~))</math>
 
| width="19%" | <math>(((~))(~))</math>
Line 183: Line 302:
 
\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>
Line 199: Line 318:
 
\end{array}</math>
 
\end{array}</math>
 
| width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
 
| width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
−
| width="19%" | <math>\cdots</math>
+
| width="19%" | [[Image:Riff 5 Big.jpg|65px]]
 
| width="19%" | [[Image:Rote 5 Big.jpg|40px]]
 
| width="19%" | [[Image:Rote 5 Big.jpg|40px]]
 
| width="19%" | <math>((((~))(~))(~))</math>
 
| width="19%" | <math>((((~))(~))(~))</math>
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\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>
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\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>
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\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>
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\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>
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\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>
Line 266: Line 385:
 
|}
 
|}
  
−
<br>
+
====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"
 +
|
 +
{| 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%" | &nbsp;
 +
| width="19%" | &nbsp;
 +
| width="19%" | [[Image:Rote 1 Big.jpg|12px]]
 +
| width="19%" | &nbsp;
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>2\!</math>
 +
| width="19%" | <math>\text{p}_1^1\!</math>
 +
| 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>
 +
|}
 +
|-
 +
|
 +
{| 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>
 +
|}
 +
|-
 +
|
 +
{| 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|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
 +
& = & \text{p}_{\text{p}_1^2}^1
 +
\\[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
 +
& = & \text{p}_1^{\text{p}_2^1}
 +
\\[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}
 +
\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===
 
===Old ASCII Version===
Line 762: Line 1,015:
 
</pre>
 
</pre>
  
−
==Example of a Raw HTML Table==
+
==A062504==
 +
 
 +
* [http://oeis.org/wiki/A062504 A062504]
  
−
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
+
===TeX Array===
  
−
<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
+
{| 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>
 +
|}
  
−
<tr>
+
===JPEG===
−
<td style="border-bottom:1px solid black"><math>\text{Mnemonic}\!</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\text{Category}\!</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\text{Classical Form}\!</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\text{Alternate Form}\!</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\text{Symmetric Form}\!</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\text{Operator}\!</math></td></tr>
 
  
−
<tr>
+
{| align="center" border="1" width="90%"
−
<td><math>\begin{matrix}
+
|+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math>
−
\mathrm{E}
+
|- 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%" | &nbsp;
 +
| width="25%" | &nbsp;
 +
| 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>&nbsp;</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>&nbsp;</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}
 
\\
 
\\
−
\mathrm{Exclusive}
+
& & & & & & & & {\color{red}2} & & 1 & & {\color{red}2}
−
\end{matrix}</math></td>
 
−
<td><math>\begin{matrix}
 
−
\mathrm{Universal}
 
 
\\
 
\\
−
\mathrm{Negative}
+
& & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3}
−
\end{matrix}</math></td>
+
\\
−
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+
& & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4}
−
<td>&nbsp;</td>
+
\\
−
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
+
& & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5}
−
<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
+
\\
 +
& & & & {\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>
 +
|}
  
−
<tr>
+
====Matrix====
−
<td style="border-bottom:1px solid black">
+
 
 +
{| align="center"
 +
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\mathrm{A}
+
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>
 +
| || &middot; ||
 +
| <font color="red">2</font>
 +
|-
 +
| || || || || || || || || || || || ||
 +
| <font color="red">3</font>
 +
| || &middot; || || &middot; ||
 +
| <font color="red">3</font>
 +
|-
 +
| || || || || || || || || || || ||
 +
| <font color="red">4</font>
 +
| || &middot; || || 2 || || &middot; ||
 +
| <font color="red">4</font>
 +
|-
 +
| || || || || || || || || || ||
 +
| <font color="red">5</font>
 +
| || &middot; || || 3 || || 1 || || &middot; ||
 +
| <font color="red">5</font>
 +
|-
 +
| || || || || || || || || ||
 +
| <font color="red">6</font>
 +
| || &middot; || || 1 || || 1 || || 4 || || &middot; ||
 +
| <font color="red">6</font>
 +
|-
 +
| || || || || || || || ||
 +
| <font color="red">7</font>
 +
| || &middot; || || 5 || || 2 || || 9 || || 1 || || &middot; ||
 +
| <font color="red">7</font>
 +
|-
 +
| || || || || || || ||
 +
| <font color="red">8</font>
 +
| || &middot; || || 6 || || 1 || || 1 || || 1 || || 2 || || &middot; ||
 +
| <font color="red">8</font>
 +
|-
 +
| || || || || || ||
 +
| <font color="red">9</font>
 +
| || &middot; || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || &middot; ||
 +
| <font color="red">9</font>
 +
|-
 +
| || || || || ||
 +
| <font color="red">10</font>
 +
| || &middot; || || 1 || || 1 || || 36|| || 1 || || 2 || || 1 || || 8 || || &middot; ||
 +
| <font color="red">10</font>
 +
|-
 +
| || || || ||
 +
| <font color="red">11</font>
 +
| || &middot; || || 1 || || 1 || || 49 || || 1 || || 5 || || 1 || || 27 || || 1 || || &middot; ||
 +
| <font color="red">11</font>
 +
|-
 +
| || || ||
 +
| <font color="red">12</font>
 +
| || &middot; || || 10 || || 3 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || &middot; ||
 +
| <font color="red">12</font>
 +
|-
 +
| || ||
 +
| <font color="red">13</font>
 +
| || &middot; || || 11 || || 1 || || 1 || || 2 || || 7 || || 1 || || 125 || || 4 || || 3 || || 1 || || &middot; ||
 +
| <font color="red">13</font>
 +
|-
 +
| ||
 +
| <font color="red">14</font>
 +
| || &middot; || || 3 || || 1 || || 100 || || 1 || || 1 || || 1 || || 216 || || 1 || || 1 || || 1 || || 4 || || &middot; ||
 +
| <font color="red">14</font>
 +
|-
 +
|
 +
| <font color="red">15</font>
 +
| || &middot; || || 13 || || 2 || || 121 || || 1 || || 3 || || 1 || || 343 || || 1 || || 5 || || 1 || || 9 || || 1 || || &middot; ||
 +
| <font color="red">15</font>
 +
|-
 +
| width="3%" | <font color="red">16</font>
 +
| width="3%" |
 +
| width="3%" | &middot;
 +
| 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%" | &middot;
 +
| 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}
 
\\
 
\\
−
\mathrm{Absolute}
+
& {\color{red}15} && \cdot && 13 && 2 && 121 && 1 && 3 && 1 && 343 && 1 && 5 && 1 && 9 && 1 && \cdot && {\color{red}15}
−
\end{matrix}</math></td>
 
−
<td style="border-bottom:1px solid black">
 
−
<math>\begin{matrix}
 
−
\mathrm{Universal}
 
 
\\
 
\\
−
\mathrm{Affirmative}
+
{\color{red}16} && \cdot && 14 && 1 && 9 && 1 && 10 && 1 && 1 && 1 && 6 && 1 && 2 && 1 && 2 && \cdot && {\color{red}16}
−
\end{matrix}</math></td>
+
\end{smallmatrix}</math>
−
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
+
|}
−
<td style="border-bottom:1px solid black">&nbsp;</td>
 
−
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
 
  
−
<tr>
+
===ASCII===
−
<td>&nbsp;</td>
 
−
<td>&nbsp;</td>
 
−
<td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td>
 
−
<td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
 
−
<td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
 
−
<td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
 
  
−
<tr>
+
<pre>
−
<td style="border-bottom:1px solid black">&nbsp;</td>
+
Example
−
<td style="border-bottom:1px solid black">&nbsp;</td>
 
−
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
 
  
−
<tr>
+
    *                                  n o m
−
<td>&nbsp;</td>
+
    *                                    \ /
−
<td>&nbsp;</td>
+
    *                                  1 . 1
−
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+
    *                                  \ / \ /
−
<td>&nbsp;</td>
+
    *                                2 .  . 2
−
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+
    *                                \ / \ / \ /
−
<td><math>\ell_{00}</math></td></tr>
+
    *                              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>
  
−
<tr>
+
==A108352==
−
<td style="border-bottom:1px solid black">&nbsp;</td>
 
−
<td style="border-bottom:1px solid black">&nbsp;</td>
 
−
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
 
−
<td style="border-bottom:1px solid black">&nbsp;</td>
 
−
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
 
−
<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
 
  
−
<tr>
+
* [http://oeis.org/wiki/A108352 A108352]
−
<td><math>\begin{matrix}
+
 
−
\mathrm{O}
+
===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.
 
\\
 
\\
−
\mathrm{Obtrusive}
+
a(8)
−
\end{matrix}</math></td>
+
& = & 2
−
<td><math>\begin{matrix}
+
& \text{because} & (\circ~ 8)^2
−
\mathrm{Particular}
+
& = & (\circ~ 1\!:\!3)^1
 +
& = & 1.
 
\\
 
\\
−
\mathrm{Negative}
+
a(9)
−
\end{matrix}</math></td>
+
& = & 0
−
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+
& \text{because} & (\circ~ 9)^k
−
<td>&nbsp;</td>
+
& = & (\circ~ 2\!:\!2)^k
−
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
+
& = & 9,
−
<td><math>\ell_{10}</math></td></tr>
+
& \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
  
−
<tr>
+
    * Table: T(n,k) = (n o)^k
−
<td><math>\begin{matrix}
+
    *                                  T(n,k)
−
\mathrm{I}
+
    *                                    \ /
 +
    *                                  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
 
\\
 
\\
−
\mathrm{Indefinite}
+
3
−
\end{matrix}</math></td>
+
&  &
−
<td><math>\begin{matrix}
+
& = & 2\!:\!1
−
\mathrm{Particular}
 
 
\\
 
\\
−
\mathrm{Affirmative}
+
2
−
\end{matrix}</math></td>
+
&  &
−
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
+
& = & 1\!:\!1
−
<td>&nbsp;</td>
+
\end{array}</math>
−
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
+
 
−
<td><math>\ell_{11}</math></td></tr>
+
: <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>
  
−
</table>
+
===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
 +
    *                                                                       
 +
    *                                                        o-o           
 +
    *                                                        |             
 +
    *                            o-o            o-o        o-o           
 +
    *                            |              |          |             
 +
    *              o-o          o-o          o-o          o-o           
 +
    *              |            |            |            |             
 +
    * O            O            O            O            O             
 +
    *                                                                       
 +
    * { }          1:1          2:1          1:2          3:1           
 +
    *                                                                       
 +
    * 1            2            3            4            5             
 +
    *                                                                       
 +
    *                                                                       
 +
    *                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 2:1      4:1          1:3          2:2          1:1 3:1       
 +
    *                                                                       
 +
    * 6            7            8            9            10           
 +
    *                                                                       
 +
    *                                                                       
 +
    * 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>
 +
 
 +
==A111795==
 +
 
 +
* [http://oeis.org/wiki/A111795 A111795]
 +
 
 +
===JPEG===
 +
 
 +
{| align="center" border="1" cellpadding="10"
 +
| valign="bottom" |
 +
<p>[[Image:Rooted Node Big.jpg|20px]]</p><br>
 +
<p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 2 Big.jpg|40px]]</p><br>
 +
<p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
 +
<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
 +
<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 5 Big.jpg|40px]]</p><br>
 +
<p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
 +
|-
 +
| valign="bottom" |
 +
<p>[[Image:Rote 7 Big.jpg|65px]]</p><br>
 +
<p><math>\begin{array}{l} 4\!:\!1 \\ 7 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 8 Big.jpg|65px]]</p><br>
 +
<p><math>\begin{array}{l} 1\!:\!3 \\ 8 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 11 Big.jpg|40px]]</p><br>
 +
<p><math>\begin{array}{l} 5\!:\!1 \\ 11 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 16 Big.jpg|90px]]</p><br>
 +
<p><math>\begin{array}{l} 1\!:\!4 \\ 16 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 17 Big.jpg|65px]]</p><br>
 +
<p><math>\begin{array}{l} 7\!:\!1 \\ 17 \end{array}</math></p>
 +
|-
 +
| valign="bottom" |
 +
<p>[[Image:Rote 19 Big.jpg|65px]]</p><br>
 +
<p><math>\begin{array}{l} 8\!:\!1 \\ 19 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 31 Big.jpg|40px]]</p><br>
 +
<p><math>\begin{array}{l} 11\!:\!1 \\ 31 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 32 Big.jpg|65px]]</p><br>
 +
<p><math>\begin{array}{l} 1\!:\!5 \\ 32 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 53 Big.jpg|90px]]</p><br>
 +
<p><math>\begin{array}{l} 16\!:\!1 \\ 53 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Rote 59 Big.jpg|65px]]</p><br>
 +
<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p>
 +
|}
 +
 
 +
===ASCII===
 +
 
 +
<pre>
 +
Example
 +
 
 +
    * Tables of Rotes and Primal Codes for a(1) to a(9)
 +
    *                                                             
 +
    *                                                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  2:1  1:2    3:1  4:1    1:3    5:1  1:4   
 +
    *                                                             
 +
    * 1    2    3    4      5    7      8      11    16   
 +
    *                                                             
 +
</pre>
 +
 
 +
==A111800==
 +
 
 +
* [http://oeis.org/wiki/A111800 A111800]
 +
 
 +
===TeX + JPEG===
 +
 
 +
<math>\text{Writing}~ \operatorname{prime}(i)^j ~\text{as}~ i\!:\!j, 2500 = 4 \cdot 625 = 2^2 5^4 = 1\!:\!2 ~~ 3\!:\!4 ~\text{has the following rote:}</math>
 +
 
 +
{| align="center" cellpadding="6"
 +
| [[Image:Rote 2500 Big.jpg|170px]]
 +
|}
 +
 
 +
<math>\text{So}~ a(2500) = a(1\!:\!2 ~~ 3\!:\!4) = a(1) + a(2) + a(3) + a(4) + 1 = 1 + 3 + 5 + 5 + 1 = 15.</math>
 +
 
 +
===ASCII===
 +
 
 +
<pre>
 +
Example
 +
 
 +
    * Writing prime(i)^j as i:j and using equal signs between identified nodes:
 +
    * 2500 = 4 * 625 = 2^2 5^4 = 1:2 3:4 has the following rote:
 +
    *               
 +
    *      o-o  o-o
 +
    *      |    | 
 +
    *  o-o o-o o-o 
 +
    *  |  |  |   
 +
    * o-o  o---o   
 +
    * |    |       
 +
    * O=====O       
 +
    *               
 +
    * So a(2500) = a(1:2 3:4) = a(1)+a(2)+a(3)+a(4)+1 = 1+3+5+5+1 = 15.
 +
</pre>

Latest revision as of 18:48, 31 January 2010

A061396

Plain Wiki Table

Large Scale

\(\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!\)
\(\text{Integer}\!\) \(\text{Factorization}\!\) \(\text{Notation}\!\) \(\text{Riff Digraph}\!\) \(\text{Rote Graph}\!\) \(\text{Traversal}\!\)
\(1\!\) \(1\!\)     Rote 1 Big.jpg  
\(2\!\) \(\text{p}_1^1\!\) \(\text{p}\!\) Riff 2 Big.jpg Rote 2 Big.jpg \(((~))\)
\(3\!\)

\(\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}\)

\(\text{p}_\text{p}\!\) Riff 3 Big.jpg Rote 3 Big.jpg \((((~))(~))\)
\(4\!\)

\(\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}\)

\(\text{p}^\text{p}\!\) Riff 4 Big.jpg Rote 4 Big.jpg \(((((~))))\)
\(5\!\) \(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \'"`UNIQ-MathJax1-QINU`"' '"`UNIQ-MathJax2-QINU`"' '"`UNIQ-MathJax3-QINU`"' '"`UNIQ-MathJax4-QINU`"' :{| border="1" cellpadding="20" | [[Image:Rote 802701 Big.jpg|330px]] |} '"`UNIQ-MathJax5-QINU`"' ==='"`UNIQ--h-40--QINU`"'JPEG=== {| align="center" border="1" cellpadding="6" | valign="bottom" | <p>[[Image:Rote 1 Big.jpg|20px]]</p><br> <p>\(1\!\)


\(a(1) ~=~ 0\)

Rote 2 Big.jpg


\(\text{p}\!\)


\(a(2) ~=~ 1\)

Rote 3 Big.jpg


\(\text{p}_\text{p}\!\)


\(a(3) ~=~ 2\)

Rote 4 Big.jpg


\(\text{p}^\text{p}\!\)


\(a(4) ~=~ 2\)

Rote 5 Big.jpg


\(\text{p}_{\text{p}_\text{p}}\!\)


\(a(5) ~=~ 3\)

Rote 6 Big.jpg


\(\text{p} \text{p}_\text{p}\!\)


\(a(6) ~=~ 2\)

Rote 7 Big.jpg


\(\text{p}_{\text{p}^\text{p}}\!\)


\(a(7) ~=~ 3\)

Rote 8 Big.jpg


\(\text{p}^{\text{p}_\text{p}}\!\)


\(a(8) ~=~ 3\)

Rote 9 Big.jpg


\(\text{p}_\text{p}^\text{p}\!\)


\(a(9) ~=~ 2\)

Rote 10 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(10) ~=~ 3\)

Rote 11 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(11) ~=~ 4\)

Rote 12 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p}\!\)


\(a(12) ~=~ 2\)

Rote 13 Big.jpg


\(\text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(a(13) ~=~ 3\)

Rote 14 Big.jpg


\(\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(a(14) ~=~ 3\)

Rote 15 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(15) ~=~ 3\)

Rote 16 Big.jpg


\(\text{p}^{\text{p}^\text{p}}\!\)


\(a(16) ~=~ 3\)

Rote 17 Big.jpg


\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(a(17) ~=~ 4\)

Rote 18 Big.jpg


\(\text{p} \text{p}_\text{p}^\text{p}\!\)


\(a(18) ~=~ 2\)

Rote 19 Big.jpg


\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(a(19) ~=~ 4\)

Rote 20 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(20) ~=~ 3\)

Rote 21 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(a(21) ~=~ 3\)

Rote 22 Big.jpg


\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(22) ~=~ 4\)

Rote 23 Big.jpg


\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(a(23) ~=~ 3\)

Rote 24 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\)


\(a(24) ~=~ 3\)

Rote 25 Big.jpg


\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(a(25) ~=~ 3\)

Rote 26 Big.jpg


\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(a(26) ~=~ 3\)

Rote 27 Big.jpg


\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(a(27) ~=~ 3\)

Rote 28 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(a(28) ~=~ 3\)

Rote 29 Big.jpg


\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(a(29) ~=~ 4\)

Rote 30 Big.jpg


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(30) ~=~ 3\)

Rote 31 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\)


\(a(31) ~=~ 5\)

Rote 32 Big.jpg


\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(32) ~=~ 4\)

Rote 33 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(33) ~=~ 4\)

Rote 34 Big.jpg


\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(a(34) ~=~ 4\)

Rote 35 Big.jpg


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(a(35) ~=~ 3\)

Rote 36 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\)


\(a(36) ~=~ 2\)

Rote 37 Big.jpg


\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\)


\(a(37) ~=~ 3\)

Rote 38 Big.jpg


\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(a(38) ~=~ 4\)

Rote 39 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(a(39) ~=~ 3\)

Rote 40 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\)


\(a(40) ~=~ 3\)

Rote 41 Big.jpg


\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\)


\(a(41) ~=~ 4\)

Rote 42 Big.jpg


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(a(42) ~=~ 3\)

Rote 43 Big.jpg


\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\)


\(a(43) ~=~ 4\)

Rote 44 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(44) ~=~ 4\)

Rote 45 Big.jpg


\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(45) ~=~ 3\)

Rote 46 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(a(46) ~=~ 3\)

Rote 47 Big.jpg


\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(a(47) ~=~ 4\)

Rote 48 Big.jpg


\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\)


\(a(48) ~=~ 3\)

Rote 49 Big.jpg


\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\)


\(a(49) ~=~ 3\)

Rote 50 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(a(50) ~=~ 3\)

Rote 51 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(a(51) ~=~ 4\)

Rote 52 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(a(52) ~=~ 3\)

Rote 53 Big.jpg


\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\)


\(a(53) ~=~ 4\)

Rote 54 Big.jpg


\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(a(54) ~=~ 3\)

Rote 55 Big.jpg


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(55) ~=~ 4\)

Rote 56 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(a(56) ~=~ 3\)

Rote 57 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(a(57) ~=~ 4\)

Rote 58 Big.jpg


\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(a(58) ~=~ 4\)

Rote 59 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\)


\(a(59) ~=~ 5\)

Rote 60 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(60) ~=~ 3\)

ASCII

 Comment

    * Table of Rotes and Primal Functions for Positive Integers from 1 to 40
    *                                                                        
    *                                                         o-o            
    *                                                         |              
    *                             o-o             o-o         o-o            
    *                             |               |           |              
    *               o-o           o-o           o-o           o-o            
    *               |             |             |             |              
    * O             O             O             O             O              
    *                                                                        
    * { }           1:1           2:1           1:2           3:1            
    *                                                                        
    * 1             2             3             4             5              
    *                                                                        
    *                                                                        
    *                 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 2:1       4:1           1:3           2:2           1:1 3:1        
    *                                                                        
    * 6             7             8             9             10             
    *                                                                        
    *                                                                        
    * 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.

A111795

JPEG

Rooted Node Big.jpg


\(\begin{array}{l} \varnothing \\ 1 \end{array}\)

Rote 2 Big.jpg


\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)

Rote 3 Big.jpg


\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)

Rote 4 Big.jpg


\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)

Rote 5 Big.jpg


\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)

Rote 7 Big.jpg


\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)

Rote 8 Big.jpg


\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)

Rote 11 Big.jpg


\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)

Rote 16 Big.jpg


\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)

Rote 17 Big.jpg


\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)

Rote 19 Big.jpg


\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)

Rote 31 Big.jpg


\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)

Rote 32 Big.jpg


\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)

Rote 53 Big.jpg


\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)

Rote 59 Big.jpg


\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)

ASCII

 Example

    * Tables of Rotes and Primal Codes for a(1) to a(9)
    *                                                              
    *                                                 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   2:1   1:2     3:1   4:1     1:3     5:1   1:4    
    *                                                              
    * 1     2     3     4       5     7       8       11    16     
    *

A111800

TeX + JPEG

\(\text{Writing}~ \operatorname{prime}(i)^j ~\text{as}~ i\!:\!j, 2500 = 4 \cdot 625 = 2^2 5^4 = 1\!:\!2 ~~ 3\!:\!4 ~\text{has the following rote:}\)

Rote 2500 Big.jpg

\(\text{So}~ a(2500) = a(1\!:\!2 ~~ 3\!:\!4) = a(1) + a(2) + a(3) + a(4) + 1 = 1 + 3 + 5 + 5 + 1 = 15.\)

ASCII

 Example

    * Writing prime(i)^j as i:j and using equal signs between identified nodes:
    * 2500 = 4 * 625 = 2^2 5^4 = 1:2 3:4 has the following rote:
    *                
    *       o-o   o-o
    *       |     |  
    *   o-o o-o o-o  
    *   |   |   |    
    * o-o   o---o    
    * |     |        
    * O=====O        
    *                
    * So a(2500) = a(1:2 3:4) = a(1)+a(2)+a(3)+a(4)+1 = 1+3+5+5+1 = 15.
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