Difference between revisions of "User:Jon Awbrey/SEQUENCES"
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===Plain Wiki Table=== | ===Plain Wiki Table=== | ||
| − | + | ====Large Scale==== | |
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | {| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | ||
| − | |+ style="height:24px" | <math> | + | |+ 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> | ||
| Line 21: | Line 21: | ||
| | | | ||
| | | | ||
| − | | [[Image: | + | | [[Image:Rote 1 Big.jpg|20px]] |
| | | | ||
|- | |- | ||
| Line 27: | Line 27: | ||
| <math>\text{p}_1^1\!</math> | | <math>\text{p}_1^1\!</math> | ||
| <math>\text{p}\!</math> | | <math>\text{p}\!</math> | ||
| − | | [[Image: | + | | [[Image:Riff 2 Big.jpg|20px]] |
| [[Image:Rote 2 Big.jpg|40px]] | | [[Image:Rote 2 Big.jpg|40px]] | ||
| <math>((~))</math> | | <math>((~))</math> | ||
| Line 37: | Line 37: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}_\text{p}\!</math> | | <math>\text{p}_\text{p}\!</math> | ||
| − | | | + | | [[Image:Riff 3 Big.jpg|40px]] |
| [[Image:Rote 3 Big.jpg|40px]] | | [[Image:Rote 3 Big.jpg|40px]] | ||
| <math>(((~))(~))</math> | | <math>(((~))(~))</math> | ||
| Line 47: | Line 47: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}^\text{p}\!</math> | | <math>\text{p}^\text{p}\!</math> | ||
| − | | | + | | [[Image:Riff 4 Big.jpg|40px]] |
| [[Image:Rote 4 Big.jpg|65px]] | | [[Image:Rote 4 Big.jpg|65px]] | ||
| <math>((((~))))</math> | | <math>((((~))))</math> | ||
| Line 60: | Line 60: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}_{\text{p}_{\text{p}}}\!</math> | | <math>\text{p}_{\text{p}_{\text{p}}}\!</math> | ||
| − | | | + | | [[Image:Riff 5 Big.jpg|65px]] |
| [[Image:Rote 5 Big.jpg|40px]] | | [[Image:Rote 5 Big.jpg|40px]] | ||
| <math>((((~))(~))(~))</math> | | <math>((((~))(~))(~))</math> | ||
| Line 71: | Line 71: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p} \text{p}_{\text{p}}\!</math> | | <math>\text{p} \text{p}_{\text{p}}\!</math> | ||
| − | | | + | | [[Image:Riff 6 Big.jpg|65px]] |
| [[Image:Rote 6 Big.jpg|80px]] | | [[Image:Rote 6 Big.jpg|80px]] | ||
| <math>((~))(((~))(~))</math> | | <math>((~))(((~))(~))</math> | ||
| Line 84: | Line 84: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math> | | <math>\text{p}_{\text{p}^{\text{p}}}\!</math> | ||
| − | | | + | | [[Image:Riff 7 Big.jpg|65px]] |
| [[Image:Rote 7 Big.jpg|65px]] | | [[Image:Rote 7 Big.jpg|65px]] | ||
| <math>(((((~))))(~))</math> | | <math>(((((~))))(~))</math> | ||
| Line 97: | Line 97: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math> | | <math>\text{p}^{\text{p}_{\text{p}}}\!</math> | ||
| − | | | + | | [[Image:Riff 8 Big.jpg|65px]] |
| [[Image:Rote 8 Big.jpg|65px]] | | [[Image:Rote 8 Big.jpg|65px]] | ||
| <math>(((((~))(~))))</math> | | <math>(((((~))(~))))</math> | ||
| Line 108: | Line 108: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}_\text{p}^\text{p}\!</math> | | <math>\text{p}_\text{p}^\text{p}\!</math> | ||
| − | | | + | | [[Image:Riff 9 Big.jpg|40px]] |
| [[Image:Rote 9 Big.jpg|80px]] | | [[Image:Rote 9 Big.jpg|80px]] | ||
| <math>(((~))(((~))))</math> | | <math>(((~))(((~))))</math> | ||
| Line 121: | Line 121: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math> | | <math>\text{p}^{\text{p}^{\text{p}}}\!</math> | ||
| − | | | + | | [[Image:Riff 16 Big.jpg|65px]] |
| [[Image:Rote 16 Big.jpg|90px]] | | [[Image:Rote 16 Big.jpg|90px]] | ||
| <math>((((((~))))))</math> | | <math>((((((~))))))</math> | ||
|} | |} | ||
| − | + | ====Small Scale==== | |
| − | + | {| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | |
| − | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> | |
| − | + | |- style="height:48px; background:#f0f0ff" | |
| − | + | | <math>\text{Integer}\!</math> | |
| − | {| align="center" border="1" | + | | <math>\text{Factorization}\!</math> |
| − | |+ style="height: | + | | <math>\text{Notation}\!</math> |
| − | |- style="height: | + | | <math>\text{Riff Digraph}\!</math> |
| + | | <math>\text{Rote Graph}\!</math> | ||
| + | | <math>\text{Traversal}\!</math> | ||
| + | |- style="height:48px" | ||
| + | | <math>1\!</math> | ||
| + | | <math>1\!</math> | ||
| + | | | ||
| + | | | ||
| + | | [[Image:Rote 1 Big.jpg|12px]] | ||
| + | | | ||
| + | |- | ||
| + | | <math>2\!</math> | ||
| + | | <math>\text{p}_1^1\!</math> | ||
| + | | <math>\text{p}\!</math> | ||
| + | | [[Image:Riff 2 Big.jpg|12px]] | ||
| + | | [[Image:Rote 2 Big.jpg|24px]] | ||
| + | | <math>((~))</math> | ||
| + | |- | ||
| + | | <math>3\!</math> | ||
| | | | ||
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 | |
| − | + | \end{array}</math> | |
| − | + | | <math>\text{p}_\text{p}\!</math> | |
| − | + | | [[Image:Riff 3 Big.jpg|24px]] | |
| − | + | | [[Image:Rote 3 Big.jpg|24px]] | |
| − | | | + | | <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> | ||
| | | | ||
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<math>\begin{array}{lll} | <math>\begin{array}{lll} | ||
| − | \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 | + | \text{p}_3^1 |
| + | & = & \text{p}_{\text{p}_2^1}^1 | ||
| + | \\[6pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | ||
\end{array}</math> | \end{array}</math> | ||
| − | + | | <math>\text{p}_{\text{p}_{\text{p}}}\!</math> | |
| − | | | + | | [[Image:Riff 5 Big.jpg|38px]] |
| − | + | | [[Image:Rote 5 Big.jpg|24px]] | |
| − | + | | <math>((((~))(~))(~))</math> | |
|- | |- | ||
| − | | <math> | + | | <math>6\!</math> |
| | | | ||
<math>\begin{array}{lll} | <math>\begin{array}{lll} | ||
| − | + | \text{p}_1^1 \text{p}_2^1 | |
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| − | \text{p}_1^1 \text{p}_2^1 | ||
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 | & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p} \text{p}_{\text{p}}\!</math> | | <math>\text{p} \text{p}_{\text{p}}\!</math> | ||
| − | | | + | | [[Image:Riff 6 Big.jpg|38px]] |
| − | | [[Image:Rote 6 Big.jpg| | + | | [[Image:Rote 6 Big.jpg|48px]] |
| <math>((~))(((~))(~))</math> | | <math>((~))(((~))(~))</math> | ||
|- | |- | ||
| Line 219: | Line 201: | ||
\text{p}_4^1 | \text{p}_4^1 | ||
& = & \text{p}_{\text{p}_1^2}^1 | & = & \text{p}_{\text{p}_1^2}^1 | ||
| − | \\[ | + | \\[6pt] |
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 | & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math> | | <math>\text{p}_{\text{p}^{\text{p}}}\!</math> | ||
| − | | | + | | [[Image:Riff 7 Big.jpg|38px]] |
| − | | [[Image:Rote 7 Big.jpg| | + | | [[Image:Rote 7 Big.jpg|38px]] |
| <math>(((((~))))(~))</math> | | <math>(((((~))))(~))</math> | ||
|- | |- | ||
| Line 232: | Line 214: | ||
\text{p}_1^3 | \text{p}_1^3 | ||
& = & \text{p}_1^{\text{p}_2^1} | & = & \text{p}_1^{\text{p}_2^1} | ||
| − | \\[ | + | \\[6pt] |
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} | & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math> | | <math>\text{p}^{\text{p}_{\text{p}}}\!</math> | ||
| − | | | + | | [[Image:Riff 8 Big.jpg|38px]] |
| − | | [[Image:Rote 8 Big.jpg| | + | | [[Image:Rote 8 Big.jpg|38px]] |
| <math>(((((~))(~))))</math> | | <math>(((((~))(~))))</math> | ||
|- | |- | ||
| Line 247: | Line 229: | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}_\text{p}^\text{p}\!</math> | | <math>\text{p}_\text{p}^\text{p}\!</math> | ||
| − | | | + | | [[Image:Riff 9 Big.jpg|24px]] |
| − | | [[Image:Rote 9 Big.jpg| | + | | [[Image:Rote 9 Big.jpg|48px]] |
| <math>(((~))(((~))))</math> | | <math>(((~))(((~))))</math> | ||
|- | |- | ||
| Line 256: | Line 238: | ||
\text{p}_1^4 | \text{p}_1^4 | ||
& = & \text{p}_1^{\text{p}_1^2} | & = & \text{p}_1^{\text{p}_1^2} | ||
| − | \\[ | + | \\[6pt] |
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} | ||
\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math> | | <math>\text{p}^{\text{p}^{\text{p}}}\!</math> | ||
| − | | | + | | [[Image:Riff 16 Big.jpg|38px]] |
| − | | [[Image:Rote 16 Big.jpg| | + | | [[Image:Rote 16 Big.jpg|52px]] |
| <math>((((((~))))))</math> | | <math>((((((~))))))</math> | ||
| − | |||
|} | |} | ||
| − | + | ===Nested Wiki Table=== | |
| − | |||
| − | === | ||
| − | + | ====Large 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%" | | ||
| + | | width="19%" | | ||
| + | | width="19%" | [[Image:Rote 1 Big.jpg|20px]] | ||
| + | | width="19%" | | ||
| + | |} | ||
| + | |- | ||
| | | | ||
| − | | | + | {| 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|20px]] | |
| + | | width="19%" | [[Image:Rote 2 Big.jpg|40px]] | ||
| + | | width="19%" | <math>((~))</math> | ||
| + | |} | ||
| + | |- | ||
| | | | ||
| − | | | + | {| cellpadding="12" style="text-align:center; width:100%" |
| − | | | + | | width="10%" | <math>3\!</math> |
| − | | | + | | width="19%" | |
| − | | 3 | + | <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|40px]] | |
| + | | width="19%" | [[Image:Rote 3 Big.jpg|40px]] | ||
| + | | 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> |
| − | | 4 | + | | <math>\text{p}^\text{p}\!</math> |
| − | | | + | | [[Image:Riff 4 Big.jpg|40px]] |
| + | | [[Image:Rote 4 Big.jpg|65px]] | ||
| + | | <math>((((~))))</math> | ||
| + | |} | ||
| + | |- | ||
| | | | ||
| − | + | {| cellpadding="12" style="text-align:center; width:100%" | |
| + | | width="10%" | <math>5\!</math> | ||
| + | | width="19%" | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_3^1 | ||
| + | & = & \text{p}_{\text{p}_2^1}^1 | ||
| + | \\[10pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | ||
| + | \end{array}</math> | ||
| + | | width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math> | ||
| + | | width="19%" | [[Image:Riff 5 Big.jpg|65px]] | ||
| + | | width="19%" | [[Image:Rote 5 Big.jpg|40px]] | ||
| + | | width="19%" | <math>((((~))(~))(~))</math> | ||
| + | |- | ||
| + | | <math>6\!</math> | ||
| | | | ||
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_1^1 \text{p}_2^1 | |
| − | | | + | & = & \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>((~))(((~))(~))</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|65px]] | |
| + | | [[Image:Rote 7 Big.jpg|65px]] | ||
| + | | <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|65px]] |
| − | | | + | | [[Image:Rote 8 Big.jpg|65px]] |
| − | + | | <math>(((((~))(~))))</math> | |
| + | |- | ||
| + | | <math>9\!</math> | ||
| | | | ||
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_2^2 | |
| − | + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} | |
| − | + | \end{array}</math> | |
| − | + | | <math>\text{p}_\text{p}^\text{p}\!</math> | |
| − | | | + | | [[Image:Riff 9 Big.jpg|40px]] |
| − | | | + | | [[Image:Rote 9 Big.jpg|80px]] |
| + | | <math>(((~))(((~))))</math> | ||
| + | |- | ||
| + | | <math>16\!</math> | ||
| | | | ||
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_1^4 | |
| − | + | & = & \text{p}_1^{\text{p}_1^2} | |
| − | | | + | \\[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|65px]] |
| + | | [[Image:Rote 16 Big.jpg|90px]] | ||
| + | | <math>((((((~))))))</math> | ||
| + | |} | ||
| + | |} | ||
| + | |||
| + | ====Small Scale==== | ||
| + | |||
| + | {| align="center" border="1" width="96%" | ||
| + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> | ||
| + | |- style="height:50px; background:#f0f0ff" | ||
| | | | ||
| − | | | + | {| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%" |
| − | | | + | | width="10%" | <math>\text{Integer}\!</math> |
| − | | | + | | width="19%" | <math>\text{Factorization}\!</math> |
| − | | | + | | width="14%" | <math>\text{Notation}\!</math> |
| − | | | + | | width="19%" | <math>\text{Riff Digraph}\!</math> |
| − | | | + | | width="19%" | <math>\text{Rote Graph}\!</math> |
| − | | | + | | width="19%" | <math>\text{Traversal}\!</math> |
| + | |} | ||
| + | |- | ||
| | | | ||
| − | + | {| cellpadding="12" style="text-align:center; width:100%" | |
| − | + | | width="10%" | <math>1\!</math> | |
| − | + | | width="19%" | <math>1\!</math> | |
| − | + | | width="14%" | | |
| − | + | | width="19%" | | |
| − | + | | width="19%" | [[Image:Rote 1 Big.jpg|12px]] | |
| − | + | | width="19%" | | |
| − | + | |} | |
| − | + | |- | |
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| | | | ||
| − | | | + | {| 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%" | | |
| − | | p | + | <math>\begin{array}{lll} |
| + | \text{p}_3^1 | ||
| + | & = & \text{p}_{\text{p}_2^1}^1 | ||
| + | \\[10pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | ||
| + | \end{array}</math> | ||
| + | | width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math> | ||
| + | | width="19%" | [[Image:Riff 5 Big.jpg|38px]] | ||
| + | | width="19%" | [[Image:Rote 5 Big.jpg|24px]] | ||
| + | | width="19%" | <math>((((~))(~))(~))</math> | ||
| + | |- | ||
| + | | <math>6\!</math> | ||
| | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^1 \text{p}_2^1 | ||
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p} \text{p}_{\text{p}}\!</math> | ||
| + | | [[Image:Riff 6 Big.jpg|38px]] | ||
| + | | [[Image:Rote 6 Big.jpg|48px]] | ||
| + | | <math>((~))(((~))(~))</math> | ||
| + | |- | ||
| + | | <math>7\!</math> | ||
| | | | ||
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_4^1 | |
| − | | | + | & = & \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=== | |
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| − | + | <pre> | |
| + | Illustration of initial terms of A061396 | ||
| + | Jon Awbrey (jawbrey(AT)oakland.edu) | ||
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | | integer factorization riff r.i.f.f. rote --> in parentheses | |
| − | o---------------------------------------------------------------------- | + | | k p's k nodes 2k+1 nodes |
| + | o-------------------------------------------------------------------------------- | ||
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| − | | | + | | 1 1 blank blank @ blank |
| − | |||
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| − | |||
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| − | + | o-------------------------------------------------------------------------------- | |
| − | |||
| − | o---------------------------------------------------------------------- | ||
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| − | | | + | | o---o |
| − | | | + | | | |
| − | | | + | | 2 p_1^1 p @ @ (()) |
| − | | | ||
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| − | + | o-------------------------------------------------------------------------------- | |
| − | |||
| − | o---------------------------------------------------------------------- | ||
| | | | ||
| − | | | + | | o---o |
| − | | | + | | | |
| − | | | + | | o---o |
| − | | | + | | 3 p_2^1 = | |
| + | | p_(p_1)^1 p_p @ @ ((())()) | ||
| + | | ^ | ||
| + | | \ | ||
| + | | o | ||
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| − | | | + | | o---o |
| − | | | + | | o | |
| − | + | | ^ o---o | |
| + | | 4 p_1^2 = / | | ||
| + | | p_1^p_1 p^p @ @ (((()))) | ||
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| − | + | o-------------------------------------------------------------------------------- | |
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| − | o-------------------------------------------------------------------------------- | ||
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| o---o | | o---o | ||
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| − | + | | o---o | |
| − | + | | | | |
| − | + | | 5 p_3 = o---o | |
| − | + | | p_(p_2) = | | |
| − | | o---o | + | | p_(p_(p_1)) p_(p_p) @ @ (((())())()) |
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| − | | | ||
| − | | p_(p_1) | ||
| ^ | | ^ | ||
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| o | | o | ||
| Line 552: | Line 588: | ||
| 7 p_4 = o---o | | 7 p_4 = o---o | ||
| p_(p_1^2) = | | | p_(p_1^2) = | | ||
| − | | p_(p_1^p_1) p | + | | p_(p_1^p_1) p_(p^p) @ o @ ((((())))()) |
| − | | | + | | ^ ^ |
| \ / | | \ / | ||
| o | | o | ||
| Line 562: | Line 598: | ||
| o | | | o | | ||
| 8 p_1^3 = ^ ^ o---o | | 8 p_1^3 = ^ ^ o---o | ||
| − | | p_1^p_2 = | + | | p_1^p_2 = / \ | |
| − | | p_1^p_(p_1) p | + | | p_1^p_(p_1) p^p_p @ o @ ((((())()))) |
| | | | ||
| o-o o-o | | o-o o-o | ||
| o | | | | o | | | ||
| 9 p_2^2 = ^ o---o | | 9 p_2^2 = ^ o---o | ||
| − | | p_(p_1)^2 = | + | | p_(p_1)^2 = / | |
| − | | p_(p_1)^(p_1) p | + | | p_(p_1)^(p_1) p_p^p @ @ ((())((()))) |
| − | | | + | | ^ |
| \ | | \ | ||
| o | | o | ||
| Line 578: | Line 614: | ||
| / o---o | | / o---o | ||
| o | | | o | | ||
| − | | 16 p_1^4 = | + | | 16 p_1^4 = ^ o---o |
| − | | p_1^(p_1^2) = | + | | p_1^(p_1^2) = / | |
| − | | p_1^(p_1^p_1) p | + | | p_1^(p_1^p_1) p^(p^p) @ @ (((((()))))) |
| | | | ||
o-------------------------------------------------------------------------------- | o-------------------------------------------------------------------------------- | ||
| − | + | Further Comments: | |
| + | |||
| + | Here are a couple more pages from my notes, | ||
| + | where it looks like I first arrived at the | ||
| + | generating function, and also carried out | ||
| + | some brute force enumerations of riffs. | ||
| − | + | I am going to experiment with a different way of | |
| + | transcribing indices and powers into a plaintext. | ||
| − | + | | jj | |
| − | | | + | | p< |
| − | | | + | | j / ji |
| − | + | | p< p< etc. | |
| + | | i \ ij | ||
| + | | p< | ||
| + | | ii | ||
| + | |||
| + | ------------------------------------------------------- | ||
| + | |||
| + | 1978-11-06 | ||
| + | |||
| + | Generating Function | ||
| + | |||
| + | | R(x) = 1 + x + 2x^2 + ... | ||
| | | | ||
| − | | 1 | + | | = 1 + x.x^0 (1 + x + 2x^2 + ...) |
| + | | . 1 + x.x^1 (1 + x + 2x^2 + ...) | ||
| + | | . 1 + x.x^2 (1 + x + 2x^2 + ...) | ||
| + | | . 1 + x.x^2 (1 + x + 2x^2 + ...) | ||
| + | | . ... | ||
| | | | ||
| − | + | | = 1 + x + 2x^2 + ... | |
| | | | ||
| − | | | + | | Product over (i = 0 to infinity) of (1 + x.x^i.R(x))^R_i = R(x) |
| − | + | ||
| − | | | + | ------------------------------------------------------- |
| + | |||
| + | 1978-11-10 | ||
| + | |||
| + | Brute force enumeration of R_n | ||
| + | |||
| + | | 4 p's | ||
| | | | ||
| − | + | | p | |
| + | | p< p_p p p | ||
| + | | p< p< p p_p p<_p p_p_p p_p< | ||
| + | | p< p< p< p< p< p< | ||
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| − | | | + | | p |
| − | | | + | | p< p_p p p |
| − | | | + | | p_p< p_p< p< p_p<_p p_p_p_p p_p_p< |
| − | | | + | | p p_p |
| − | | | + | | |
| | | | ||
| − | + | | p | |
| + | | p< p_p p p p p | ||
| + | | p< p< p< p< p< p< p p< | ||
| + | | p p p_p p^p p p | ||
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| − | | | + | | p p_p_p p p< |
| − | + | | p^p | |
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| | | | ||
| − | | | + | |
| − | | | + | Altogether, 20 riffs of weight 4. |
| − | | | + | |
| − | | | + | | o---------------------o---------------------o---------------------o |
| − | | | + | | | 3 | 4 | 5 | |
| − | | | + | | o---------------------o---------------------o---------------------| |
| − | | | + | | | // // 2 | 10, 3, 1, 6 | 36, 10, 2, 3, 2, 20 | |
| − | | | + | | o---------------------o---------------------o---------------------| |
| − | | | + | | | | 0^1 4^1, | | |
| − | | | + | | | | 1^1 3^1, | | |
| + | | | | 2^2, | | | ||
| + | | | | 4^1 0^1 | | | ||
| + | | o---------------------o---------------------o---------------------o | ||
| + | | | 6 | 20 | 73 | | ||
| + | | o---------------------o---------------------o---------------------o | ||
| | | | ||
| − | + | ||
| − | + | ------------------------------------------------------- | |
| − | + | ||
| − | + | Here are the number values of the riffs on 4 nodes: | |
| − | + | ||
| − | + | o---------------------------------------------------------------------- | |
| − | |||
| | | | ||
| − | | | + | | p |
| − | | | + | | p< p_p p p |
| − | | | + | | p< p< p p_p p<_p p_p_p p_p< |
| − | + | | p< p< p< p< p< p< | |
| − | |||
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| − | | | + | | 2^16 2^8 2^6 2^9 2^5 2^7 |
| − | + | | 65536 256 64 512 32 128 | |
| − | + | o---------------------------------------------------------------------- | |
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| − | o---------------------------------------------------------------------- | ||
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| p | | p | ||
| − | | p< p | + | | p< p_p p p |
| − | | | + | | p_p< p_p< p< p_p<_p p_p_p_p p_p_p< |
| − | | p | + | | p p_p |
| | | | ||
| − | | | + | | p_16 p_8 p_6 p_9 p_5 p_7 |
| − | | | + | | 53 19 13 23 11 17 |
| + | o---------------------------------------------------------------------- | ||
| | | | ||
| − | + | | p | |
| + | | p< p_p p p p | ||
| + | | p< p< p< p< p^p p_p p p< | ||
| + | | p p p_p p^p p | ||
| | | | ||
| − | | | + | | 3^4 3^3 5^2 7^2 |
| − | | | + | | 81 27 25 49 12 18 |
| − | + | o---------------------------------------------------------------------- | |
| − | |||
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| − | | | + | | p p_p_p p p< |
| − | | | + | | p^p |
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| − | o-------------------------------------------------------------------------------- | + | | 10 14 |
| + | o---------------------------------------------------------------------- | ||
| + | |||
| + | For ease of reference, I include the previous table | ||
| + | of smaller riffs and rotes, redone in the new style. | ||
| + | |||
| + | o-------------------------------------------------------------------------------- | ||
| + | | integer factorization riff r.i.f.f. rote --> in parentheses | ||
| + | | k p's k nodes 2k+1 nodes | ||
| + | o-------------------------------------------------------------------------------- | ||
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| − | | | + | | 1 1 blank blank @ blank |
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o-------------------------------------------------------------------------------- | o-------------------------------------------------------------------------------- | ||
| | | | ||
| − | | | + | | o---o |
| − | | | + | | | |
| − | | | + | | 2 p_1^1 p @ @ (()) |
| | | | ||
| − | + | o-------------------------------------------------------------------------------- | |
| | | | ||
| − | o= | + | | o---o |
| − | + | | | | |
| − | + | | o---o | |
| − | + | | 3 p_2^1 = | | |
| − | + | | p_(p_1)^1 p_p @ @ ((())()) | |
| − | + | | ^ | |
| − | -- | + | | \ |
| − | + | | o | |
| − | + | | | |
| − | + | | o---o | |
| − | + | | o | | |
| − | + | | ^ o---o | |
| − | + | | 4 p_1^2 = / | | |
| − | + | | p_1^p_1 p^p @ @ (((()))) | |
| − | + | | | |
| − | + | o-------------------------------------------------------------------------------- | |
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| − | | | + | | o---o |
| − | + | | | | |
| − | + | | o---o | |
| − | + | | | | |
| − | + | | 5 p_3 = o---o | |
| − | + | | p_(p_2) = | | |
| − | + | | p_(p_(p_1)) p_p_p @ @ (((())())()) | |
| − | + | | ^ | |
| − | + | | \ | |
| − | + | | o | |
| − | + | | ^ | |
| − | + | | \ | |
| − | + | | o | |
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| − | + | | o-o | |
| − | + | | / | |
| + | | o-o o-o | ||
| + | | 6 p_1 p_2 = \ / | ||
| + | | p_1 p_(p_1) p p_p @ @ @ (())((())()) | ||
| + | | ^ | ||
| + | | \ | ||
| + | | o | ||
| | | | ||
| − | + | | o---o | |
| − | + | | | | |
| + | | o---o | ||
| + | | | | ||
| + | | 7 p_4 = o---o | ||
| + | | p_(p_1^2) = | | ||
| + | | p_(p_1^p_1) p< @ o @ ((((())))()) | ||
| + | | p^p ^ ^ | ||
| + | | \ / | ||
| + | | o | ||
| | | | ||
| − | + | | o---o | |
| − | + | | | | |
| + | | o---o | ||
| + | | o | | ||
| + | | 8 p_1^3 = ^ ^ o---o | ||
| + | | p_1^p_2 = p_p / \ | | ||
| + | | p_1^p_(p_1) p< @ o @ ((((())()))) | ||
| | | | ||
| − | + | | o-o o-o | |
| − | + | | o | | | |
| + | | 9 p_2^2 = ^ o---o | ||
| + | | p_(p_1)^2 = p / | | ||
| + | | p_(p_1)^(p_1) p< @ @ ((())((()))) | ||
| + | | p ^ | ||
| + | | \ | ||
| + | | o | ||
| | | | ||
| − | + | | o o---o | |
| − | + | | ^ | | |
| + | | / o---o | ||
| + | | o | | ||
| + | | 16 p_1^4 = p ^ o---o | ||
| + | | p_1^(p_1^2) = p< / | | ||
| + | | p_1^(p_1^p_1) p< @ @ (((((()))))) | ||
| | | | ||
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | ||
| − | + | (later) | |
| − | |||
| − | |||
| − | |||
| − | + | Expanded version of first table: | |
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | | integer factorization riff r.i.f.f. rote --> in parentheses | |
| − | + | | k p's k nodes 2k+1 nodes | |
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | | | |
| − | + | | 1 1 blank blank @ blank | |
| − | + | | | |
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | | | |
| − | + | | o---o | |
| − | + | | | | |
| − | + | | 2 p_1^1 p @ @ (()) | |
| − | + | | | |
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | | | |
| − | + | | o---o | |
| − | + | | | | |
| − | + | | o---o | |
| − | + | | 3 p_2^1 = | | |
| − | + | | p_(p_1)^1 p_p @ @ ((())()) | |
| − | + | | ^ | |
| − | + | | \ | |
| − | + | | o | |
| − | + | | | |
| − | + | | o---o | |
| − | + | | o | | |
| − | + | | ^ o---o | |
| − | + | | 4 p_1^2 = / | | |
| − | + | | p_1^p_1 p^p @ @ (((()))) | |
| − | + | | | |
| − | + | o-------------------------------------------------------------------------------- | |
| − | | | + | | |
| − | + | | o---o | |
| − | + | | | | |
| − | + | | o---o | |
| − | + | | | | |
| − | + | | 5 p_3 = o---o | |
| − | + | | p_(p_2) = | | |
| − | + | | p_(p_(p_1)) p_p_p @ @ (((())())()) | |
| − | + | | ^ | |
| − | | | + | | \ |
| − | + | | o | |
| − | + | | ^ | |
| − | | | + | | \ |
| − | | | + | | o |
| − | + | | | |
| − | + | | o-o | |
| − | | | + | | / |
| − | + | | o-o o-o | |
| − | + | | 6 p_1 p_2 = \ / | |
| − | + | | p_1 p_(p_1) p p_p @ @ @ (())((())()) | |
| − | + | | ^ | |
| − | + | | \ | |
| − | + | | o | |
| − | + | | | |
| − | + | | o---o | |
| − | | | + | | | |
| − | + | | o---o | |
| − | + | | | | |
| − | + | | 7 p_4 = o---o | |
| − | | | + | | p_(p_1^2) = | |
| − | + | | p_(p_1^p_1) p< @ o @ ((((())))()) | |
| − | + | | p^p ^ ^ | |
| − | | | + | | \ / |
| − | + | | o | |
| − | + | | | |
| − | | | + | | o---o |
| − | + | | | | |
| − | + | | o---o | |
| − | | | + | | o | |
| − | + | | 8 p_1^3 = ^ ^ o---o | |
| − | + | | p_1^p_2 = p_p / \ | | |
| − | | | + | | p_1^p_(p_1) p< @ o @ ((((())()))) |
| − | + | | | |
| − | + | | o-o o-o | |
| − | + | | o | | | |
| − | | | + | | 9 p_2^2 = ^ o---o |
| − | < | + | | p_(p_1)^2 = p / | |
| − | + | | p_(p_1)^(p_1) p< @ @ ((())((()))) | |
| − | | | + | | p ^ |
| − | + | | \ | |
| − | + | | o | |
| − | | | + | | |
| − | + | | o o---o | |
| − | + | | ^ | | |
| − | | | + | | / o---o |
| − | + | | o | | |
| − | + | | 16 p_1^4 = p ^ o---o | |
| − | + | | p_1^(p_1^2) = p< / | | |
| − | + | | p_1^(p_1^p_1) p< @ @ (((((()))))) | |
| − | + | | | |
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | ||
| − | + | o================================================================================ | |
| − | + | | | |
| − | + | | p | |
| − | + | | p< p p_p p | |
| − | + | | p< p<_p p< p_p< p p_p p_p_p | |
| − | + | | p< p< p< p< p< p< | |
| − | + | | | |
| − | + | | 2^16 2^9 2^8 2^7 2^6 2^5 | |
| − | + | | 65536 512 256 128 64 32 | |
| − | + | | | |
| − | <p | + | o-------------------------------------------------------------------------------- |
| − | + | | | |
| − | <p | + | | p |
| − | <p | + | | p< p p_p p |
| − | | | + | | p_p< p_p<_p p_p< p_p_p< p< p_p_p_p |
| − | | | + | | p p_p |
| − | + | | | |
| − | + | | p_16 p_9 p_8 p_7 p_6 p_5 | |
| − | + | | 53 23 19 17 13 11 | |
| − | + | | | |
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | | | |
| − | + | | p^p p_p p p | |
| − | < | + | | p< p< p< p< |
| − | | | + | | p p p^p p_p |
| − | + | | | |
| − | + | | 3^4 3^3 7^2 5^2 | |
| − | + | | 81 27 49 25 | |
| − | + | | | |
| − | + | o-------------------------------------------------------------------------------- | |
| − | + | | | |
| − | + | | p | |
| − | + | | p p< p p< p^p p_p p p_p_p | |
| − | <p | + | | p p^p |
| − | | | + | | |
| − | + | | 18 14 12 10 | |
| − | + | | | |
| − | + | o================================================================================ | |
| − | + | ||
| − | + | Triangle in which k-th row lists natural number | |
| − | + | values for the collection of riffs with k nodes. | |
| − | + | ||
| − | + | k | natural numbers n such that |riff(n)| = k | |
| − | | | + | --o------------------------------------------------ |
| − | + | 0 | 1; | |
| − | + | 1 | 2; | |
| − | + | 2 | 3, 4; | |
| − | + | 3 | 5, 6, 7, 8, 9, 16; | |
| − | + | 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, | |
| − | + | | 32, 49, 53, 64, 81, 128, 256, 512, 65536; | |
| − | + | ||
| − | + | The natural number values for the riffs with | |
| − | + | at most 3 pts are as follows (@'s are roots): | |
| − | + | ||
| − | + | | o o o o | |
| − | + | | | ^ | ^ | |
| − | + | | v | v | | |
| − | + | | o o o o o o o o o | |
| − | + | | | ^ | | | ^ | ^ ^ | |
| − | + | | v | v v v | v/ | | |
| − | + | | Riff: @; @, @; @, @ @, @, @, @, @; | |
| − | + | | | |
| − | | | + | | Value: 2; 3, 4; 5, 6 , 7, 8, 9, 16; |
| − | | | + | |
| − | + | --------------------------------------------------- | |
| − | + | ||
| − | + | 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, | |
| − | + | 10, 11, 12, 13, 14, 17, 18, 19, | |
| − | + | 23, 25, 27, 32, 49, 53, 64, 81, | |
| − | + | 128, 256, 512, 65536, | |
| − | + | ||
| − | + | --------------------------------------------------- | |
| − | | | + | |
| − | + | 1; 2; 3, 4; 5, 6, 7, 8, 9, 16; | |
| − | + | 10, 11, 12, 13, 14, 17, 18, 19, | |
| − | + | 23, 25, 27, 32, 49, 53, 64, 81, | |
| − | + | 128, 256, 512, 65536; | |
| − | + | ||
| − | + | --------------------------------------------------- | |
| − | + | </pre> | |
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| − | + | ==A062504== | |
| − | + | * [http://oeis.org/wiki/A062504 A062504] | |
| − | + | ===TeX Array=== | |
| − | |||
| − | + | {| align="center" | |
| − | + | | | |
| − | + | <math>\begin{array}{l|l|r} | |
| − | + | k | |
| − | + | & P_k | |
| − | + | = \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \} | |
| − | + | = \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \} | |
| − | + | & |P_k| | |
| − | + | \\[10pt] | |
| − | + | 0 & \{ 1 \} & 1 | |
| − | + | \\ | |
| − | + | 1 & \{ 2 \} & 1 | |
| − | + | \\ | |
| − | + | 2 & \{ 3, 4 \} & 2 | |
| − | + | \\ | |
| − | + | 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 | |
| − | + | \\ | |
| − | + | 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 | |
| − | + | \end{array}</math> | |
| − | + | |} | |
| − | + | ||
| − | + | ===JPEG=== | |
| − | + | ||
| − | + | {| align="center" border="1" width="90%" | |
| − | + | |+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math> | |
| − | + | |- style="height:50px; background:#f0f0ff" | |
| − | + | | | |
| − | + | {| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%" | |
| − | + | | width="10%" | <math>\text{Integer}\!</math> | |
| − | + | | width="25%" | <math>\text{Factorization}\!</math> | |
| − | + | | width="15%" | <math>\text{Notation}\!</math> | |
| − | + | | width="25%" | <math>\text{Riff Digraph}\!</math> | |
| − | + | | width="25%" | <math>\text{Rote Graph}\!</math> | |
| − | + | |} | |
| − | + | |- | |
| − | + | | | |
| − | + | {| cellpadding="12" style="text-align:center; width:100%" | |
| − | + | | width="10%" | <math>1\!</math> | |
| − | + | | width="25%" | <math>1\!</math> | |
| − | + | | width="15%" | | |
| − | + | | width="25%" | | |
| − | + | | width="25%" | [[Image:Rote 1 Big.jpg|20px]] | |
| − | + | |} | |
| − | + | |- | |
| − | + | | | |
| − | + | {| cellpadding="12" style="text-align:center; width:100%" | |
| − | + | | width="10%" | <math>2\!</math> | |
| − | + | | width="25%" | <math>\text{p}_1^1\!</math> | |
| − | + | | width="15%" | <math>\text{p}\!</math> | |
| − | + | | width="25%" | [[Image:Riff 2 Big.jpg|20px]] | |
| − | + | | width="25%" | [[Image:Rote 2 Big.jpg|40px]] | |
| − | + | |} | |
| − | + | |- | |
| − | + | | | |
| − | + | {| cellpadding="12" style="text-align:center; width:100%" | |
| − | + | | width="10%" | <math>3\!</math> | |
| − | + | | width="25%" | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 | |
| − | + | \end{array}</math> | |
| − | + | | width="15%" | <math>\text{p}_\text{p}\!</math> | |
| − | + | | width="25%" | [[Image:Riff 3 Big.jpg|40px]] | |
| − | + | | width="25%" | [[Image:Rote 3 Big.jpg|40px]] | |
| − | + | |- | |
| − | + | | <math>4\!</math> | |
| − | + | | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} | |
| − | + | \end{array}</math> | |
| − | + | | <math>\text{p}^\text{p}\!</math> | |
| − | + | | [[Image:Riff 4 Big.jpg|40px]] | |
| − | + | | [[Image:Rote 4 Big.jpg|65px]] | |
| − | + | |} | |
| − | + | |- | |
| − | + | | | |
| − | + | {| cellpadding="12" style="text-align:center; width:100%" | |
| − | + | | width="10%" | <math>5\!</math> | |
| − | + | | width="25%" | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_3^1 | |
| − | + | & = & \text{p}_{\text{p}_2^1}^1 | |
| − | + | \\[12pt] | |
| − | + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | |
| − | + | \end{array}</math> | |
| − | + | | width="15%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math> | |
| − | + | | width="25%" | [[Image:Riff 5 Big.jpg|65px]] | |
| − | + | | width="25%" | [[Image:Rote 5 Big.jpg|40px]] | |
| − | + | |- | |
| − | + | | <math>6\!</math> | |
| − | + | | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_1^1 \text{p}_2^1 | |
| − | + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 | |
| − | + | \end{array}</math> | |
| − | + | | <math>\text{p} \text{p}_{\text{p}}\!</math> | |
| − | + | | [[Image:Riff 6 Big.jpg|65px]] | |
| − | + | | [[Image:Rote 6 Big.jpg|80px]] | |
| − | + | |- | |
| − | + | | <math>7\!</math> | |
| − | + | | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_4^1 | |
| − | + | & = & \text{p}_{\text{p}_1^2}^1 | |
| − | + | \\[12pt] | |
| − | + | & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 | |
| − | + | \end{array}</math> | |
| − | + | | <math>\text{p}_{\text{p}^{\text{p}}}\!</math> | |
| − | + | | [[Image:Riff 7 Big.jpg|65px]] | |
| − | + | | [[Image:Rote 7 Big.jpg|65px]] | |
| − | + | |- | |
| − | + | | <math>8\!</math> | |
| − | + | | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_1^3 | |
| − | + | & = & \text{p}_1^{\text{p}_2^1} | |
| − | + | \\[12pt] | |
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}_{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 8 Big.jpg|65px]] | ||
| + | | [[Image:Rote 8 Big.jpg|65px]] | ||
| + | |- | ||
| + | | <math>9\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_2^2 | ||
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_\text{p}^\text{p}\!</math> | ||
| + | | [[Image:Riff 9 Big.jpg|40px]] | ||
| + | | [[Image:Rote 9 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>16\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^4 | ||
| + | & = & \text{p}_1^{\text{p}_1^2} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}^{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 16 Big.jpg|65px]] | ||
| + | | [[Image:Rote 16 Big.jpg|90px]] | ||
| + | |} | ||
| + | |- | ||
| + | | | ||
| + | {| cellpadding="12" style="text-align:center; width:100%" | ||
| + | | width="10%" | <math>10\!</math> | ||
| + | | width="25%" | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^1 \text{p}_3^1 | ||
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | ||
| + | \end{array}</math> | ||
| + | | width="15%" | <math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math> | ||
| + | | width="25%" | [[Image:Riff 10 Big.jpg|90px]] | ||
| + | | width="25%" | [[Image:Rote 10 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>11\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_5^1 | ||
| + | & = & \text{p}_{\text{p}_3^1}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math> | ||
| + | | [[Image:Riff 11 Big.jpg|90px]] | ||
| + | | [[Image:Rote 11 Big.jpg|40px]] | ||
| + | |- | ||
| + | | <math>12\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^2 \text{p}_2^1 | ||
| + | & = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}} \text{p}_{\text{p}}\!</math> | ||
| + | | [[Image:Riff 12 Big.jpg|65px]] | ||
| + | | [[Image:Rote 12 Big.jpg|105px]] | ||
| + | |- | ||
| + | | <math>13\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_6^1 | ||
| + | & = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 13 Big.jpg|65px]] | ||
| + | | [[Image:Rote 13 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>14\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^1 \text{p}_4^1 | ||
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 14 Big.jpg|90px]] | ||
| + | | [[Image:Rote 14 Big.jpg|105px]] | ||
| + | |- | ||
| + | | <math>17\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_7^1 | ||
| + | & = & \text{p}_{\text{p}_4^1}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 17 Big.jpg|90px]] | ||
| + | | [[Image:Rote 17 Big.jpg|65px]] | ||
| + | |- | ||
| + | | <math>18\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^1 \text{p}_2^2 | ||
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math> | ||
| + | | [[Image:Riff 18 Big.jpg|65px]] | ||
| + | | [[Image:Rote 18 Big.jpg|120px]] | ||
| + | |- | ||
| + | | <math>19\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_8^1 | ||
| + | & = & \text{p}_{\text{p}_1^3}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 19 Big.jpg|90px]] | ||
| + | | [[Image:Rote 19 Big.jpg|65px]] | ||
| + | |- | ||
| + | | <math>23\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_9^1 | ||
| + | & = & \text{p}_{\text{p}_2^2}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 23 Big.jpg|65px]] | ||
| + | | [[Image:Rote 23 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>25\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_3^2 | ||
| + | & = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math> | ||
| + | | [[Image:Riff 25 Big.jpg|65px]] | ||
| + | | [[Image:Rote 25 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>27\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_2^3 | ||
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 27 Big.jpg|65px]] | ||
| + | | [[Image:Rote 27 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>32\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^5 | ||
| + | & = & \text{p}_1^{\text{p}_3^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 32 Big.jpg|90px]] | ||
| + | | [[Image:Rote 32 Big.jpg|65px]] | ||
| + | |- | ||
| + | | <math>49\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_4^2 | ||
| + | & = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math> | ||
| + | | [[Image:Riff 49 Big.jpg|65px]] | ||
| + | | [[Image:Rote 49 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>53\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_{16}^1 | ||
| + | & = & \text{p}_{\text{p}_1^4}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 53 Big.jpg|90px]] | ||
| + | | [[Image:Rote 53 Big.jpg|90px]] | ||
| + | |- | ||
| + | | <math>64\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^6 | ||
| + | & = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 64 Big.jpg|65px]] | ||
| + | | [[Image:Rote 64 Big.jpg|105px]] | ||
| + | |- | ||
| + | | <math>81\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_2^4 | ||
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 81 Big.jpg|65px]] | ||
| + | | [[Image:Rote 81 Big.jpg|105px]] | ||
| + | |- | ||
| + | | <math>128\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^7 | ||
| + | & = & \text{p}_1^{\text{p}_4^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 128 Big.jpg|90px]] | ||
| + | | [[Image:Rote 128 Big.jpg|90px]] | ||
| + | |- | ||
| + | | <math>256\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^8 | ||
| + | & = & \text{p}_1^{\text{p}_1^3} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 256 Big.jpg|90px]] | ||
| + | | [[Image:Rote 256 Big.jpg|90px]] | ||
| + | |- | ||
| + | | <math>512\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^9 | ||
| + | & = & \text{p}_1^{\text{p}_2^2} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 512 Big.jpg|65px]] | ||
| + | | [[Image:Rote 512 Big.jpg|105px]] | ||
| + | |- | ||
| + | | <math>65536\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^{16} | ||
| + | & = & \text{p}_1^{\text{p}_1^4} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 65536 Big.jpg|90px]] | ||
| + | | [[Image:Rote 65536 Big.jpg|115px]] | ||
| + | |} | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| + | |||
| + | <pre> | ||
| + | Example | ||
| + | |||
| + | * k | natural numbers n such that |riff(n)| = k | ||
| + | * 0 | 1; | ||
| + | * 1 | 2; | ||
| + | * 2 | 3, 4; | ||
| + | * 3 | 5, 6, 7, 8, 9, 16; | ||
| + | * 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536; | ||
| + | * The natural number values for the riffs with at most 3 pts are as follows (x = root): | ||
| + | * .................o.......o..o.......o | ||
| + | * .................|.......^..|.......^ | ||
| + | * .................v.......|..v.......| | ||
| + | * ...........o..o..o....o..o..o..o.o..o | ||
| + | * ...........|..^..|....|..|..^..|.^..^ | ||
| + | * ...........v..|..v....v..v..|..v/...| | ||
| + | * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x; | ||
| + | * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16; | ||
| + | </pre> | ||
| + | |||
| + | ==A062537== | ||
| + | |||
| + | * [http://oeis.org/wiki/A062537 A062537] | ||
| + | |||
| + | ===Wiki + TeX + JPEG=== | ||
| + | |||
| + | {| align="center" border="1" cellpadding="10" | ||
| + | |+ style="height:25px" | <math>a(n) = \text{Number of Nodes in the Riff of}~ n</math> | ||
| + | | valign="bottom" | | ||
| + | <p> </p><br> | ||
| + | <p><math>1\!</math></p><br> | ||
| + | <p><math>a(1) ~=~ 0</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> | ||
| + | <p><math>\text{p}\!</math></p><br> | ||
| + | <p><math>a(2) ~=~ 1</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(3) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 4 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(4) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(5) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 6 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}}\!</math></p><br> | ||
| + | <p><math>a(6) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 7 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(7) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 8 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(8) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 9 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(9) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(10) ~=~ 4</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 11 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!</math></p><br> | ||
| + | <p><math>a(11) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 12 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(12) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 13 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(13) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 14 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(14) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(15) ~=~ 5</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 16 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}^{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(16) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 17 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math></p><br> | ||
| + | <p><math>a(17) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 18 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(18) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 19 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math></p><br> | ||
| + | <p><math>a(19) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 20 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(20) ~=~ 5</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 21 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(21) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 22 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(22) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 23 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(23) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 24 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(24) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 25 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(25) ~=~ 4</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 26 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(26) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 27 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(27) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 28 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(28) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 29 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(29) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 30 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(30) ~=~ 6</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 31 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br> | ||
| + | <p><math>a(31) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 32 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(32) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 33 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(33) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 34 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(34) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 35 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(35) ~=~ 6</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 36 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(36) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 37 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(37) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 38 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(38) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 39 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(39) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 40 Big.jpg|135px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(40) ~=~ 6</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 41 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(41) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 42 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(42) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 43 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(43) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 44 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(44) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 45 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(45) ~=~ 6</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 46 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(46) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 47 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(47) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 48 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(48) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 49 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(49) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 50 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(50) ~=~ 5</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 51 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(51) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 52 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(52) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 53 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(53) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 54 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(54) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 55 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(55) ~=~ 7</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 56 Big.jpg|135px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(56) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 57 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(57) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 58 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(58) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 59 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br> | ||
| + | <p><math>a(59) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 60 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(60) ~=~ 7</math></p> | ||
| + | |} | ||
| + | |||
| + | ==A062860== | ||
| + | |||
| + | * [http://oeis.org/wiki/A062860 A062860] | ||
| + | |||
| + | ===Wiki + TeX + JPEG=== | ||
| + | |||
| + | {| align="center" border="1" cellpadding="10" | ||
| + | |+ style="height:25px" | <math>a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}</math> | ||
| + | | valign="bottom" | | ||
| + | <p> </p><br> | ||
| + | <p><math>1\!</math></p><br> | ||
| + | <p><math>a(0) ~=~ 1</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> | ||
| + | <p><math>\text{p}\!</math></p><br> | ||
| + | <p><math>a(1) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(2) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(3) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(4) ~=~ 10</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(5) ~=~ 15</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 30 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(6) ~=~ 30</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 55 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(7) ~=~ 55</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 105 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(8) ~=~ 105</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 165 Big.jpg|135px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(9) ~=~ 165</math></p> | ||
| + | |} | ||
| + | |||
| + | ==A106177== | ||
| + | |||
| + | * [http://oeis.org/wiki/A106177 A106177] | ||
| + | |||
| + | ===Primal Codes of Finite Partial Functions on Positive Integers=== | ||
| + | |||
| + | {| align="center" | ||
| + | | | ||
| + | <math>\begin{array}{rcl} | ||
| + | 1 & = & \varnothing \\ | ||
| + | 2 & = & 1\!:\!1 \\ | ||
| + | 3 & = & 2\!:\!1 \\ | ||
| + | 4 & = & 1\!:\!2 \\ | ||
| + | 5 & = & 3\!:\!1 \\ | ||
| + | 6 & = & 1\!:\!1 ~~ 2\!:\!1 \\ | ||
| + | 7 & = & 4\!:\!1 \\ | ||
| + | 8 & = & 1\!:\!3 \\ | ||
| + | 9 & = & 2\!:\!2 \\ | ||
| + | 10 & = & 1\!:\!1 ~~ 3\!:\!1 \\ | ||
| + | 11 & = & 5\!:\!1 \\ | ||
| + | 12 & = & 1\!:\!2 ~~ 2\!:\!1 \\ | ||
| + | 13 & = & 6\!:\!1 \\ | ||
| + | 14 & = & 1\!:\!1 ~~ 4\!:\!1 \\ | ||
| + | 15 & = & 2\!:\!1 ~~ 3\!:\!1 \\ | ||
| + | 16 & = & 1\!:\!4 \\ | ||
| + | 17 & = & 7\!:\!1 \\ | ||
| + | 18 & = & 1\!:\!1 ~~ 2\!:\!2 \\ | ||
| + | 19 & = & 8\!:\!1 \\ | ||
| + | 20 & = & 1\!:\!2 ~~ 3\!:\!1 | ||
| + | \end{array}</math> | ||
| + | |} | ||
| + | |||
| + | ===Wiki Table=== | ||
| + | |||
| + | {| align="center" style="font-weight:bold; text-align:center" | ||
| + | | || || || || || || || || | ||
| + | | <font color="red">1</font> | ||
| + | | | ||
| + | | <font color="red">1</font> | ||
| + | |- | ||
| + | | || || || || || || || | ||
| + | | <font color="red">2</font> | ||
| + | | || 1 || | ||
| + | | <font color="red">2</font> | ||
| + | |- | ||
| + | | || || || || || || | ||
| + | | <font color="red">3</font> | ||
| + | | || 1 || || 1 || | ||
| + | | <font color="red">3</font> | ||
| + | |- | ||
| + | | || || || || || | ||
| + | | <font color="red">4</font> | ||
| + | | || 1 || || 2 || || 1 || | ||
| + | | <font color="red">4</font> | ||
| + | |- | ||
| + | | || || || || | ||
| + | | <font color="red">5</font> | ||
| + | | || 1 || || 3 || || 1 || || 1 || | ||
| + | | <font color="red">5</font> | ||
| + | |- | ||
| + | | || || || | ||
| + | | <font color="red">6</font> | ||
| + | | || 1 || || 1 || || 1 || || 4 || || 1 || | ||
| + | | <font color="red">6</font> | ||
| + | |- | ||
| + | | || || | ||
| + | | <font color="red">7</font> | ||
| + | | || 1 || || 5 || || 2 || || 9 || || 1 || || 1 || | ||
| + | | <font color="red">7</font> | ||
| + | |- | ||
| + | | || | ||
| + | | <font color="red">8</font> | ||
| + | | || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">8</font> | ||
| + | |- | ||
| + | | | ||
| + | | <font color="red">9</font> | ||
| + | | || 1 || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || 1 || | ||
| + | | <font color="red">9</font> | ||
| + | |- | ||
| + | | width="12pt" | <font color="red">10</font> | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 36 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 2 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 8 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | <font color="red">10</font> | ||
| + | |} | ||
| + | |||
| + | ===Wiki + TeX=== | ||
| + | |||
| + | ====Smallmatrix==== | ||
| + | |||
| + | {| align="center" | ||
| + | | | ||
| + | <math>\begin{smallmatrix} | ||
| + | & & & & & & & & & {\color{red}1} & & {\color{red}1} | ||
| + | \\ | ||
| + | & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} | ||
| + | \\ | ||
| + | & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} | ||
| + | \\ | ||
| + | & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} | ||
| + | \\ | ||
| + | & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} | ||
| + | \\ | ||
| + | & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} | ||
| + | \\ | ||
| + | & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} | ||
| + | \\ | ||
| + | & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} | ||
| + | \\ | ||
| + | & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} | ||
| + | \\ | ||
| + | {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} | ||
| + | \end{smallmatrix}</math> | ||
| + | |} | ||
| + | |||
| + | ====Array==== | ||
| + | |||
| + | {| align="center" | ||
| + | | | ||
| + | <math>\begin{array}{*{21}{c}} | ||
| + | & & & & & & & & & {\color{red}1} & & {\color{red}1} | ||
| + | \\ | ||
| + | & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} | ||
| + | \\ | ||
| + | & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} | ||
| + | \\ | ||
| + | & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} | ||
| + | \\ | ||
| + | & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} | ||
| + | \\ | ||
| + | & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} | ||
| + | \\ | ||
| + | & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} | ||
| + | \\ | ||
| + | & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} | ||
| + | \\ | ||
| + | & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} | ||
| + | \\ | ||
| + | {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} | ||
| + | \end{array}</math> | ||
| + | |} | ||
| + | |||
| + | ====Matrix==== | ||
| + | |||
| + | {| align="center" | ||
| + | | | ||
| + | <math>\begin{matrix} | ||
| + | n \circ m | ||
| + | \\ | ||
| + | 1 ~/~\backslash~ 1 | ||
| + | \\ | ||
| + | 2 ~/~ 1 ~\backslash~ 2 | ||
| + | \\ | ||
| + | 3 ~/~ 1 \cdot 1 ~\backslash~ 3 | ||
| + | \\ | ||
| + | 4 ~/~ 1 \cdot 2 \cdot 1 ~\backslash~ 4 | ||
| + | \\ | ||
| + | 5 ~/~ 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 5 | ||
| + | \\ | ||
| + | 6 ~/~ 1 \cdot 1 \cdot 1 \cdot 4 \cdot 1 ~\backslash~ 6 | ||
| + | \\ | ||
| + | 7 ~/~ 1 \cdot 5 \cdot 2 \cdot 9 \cdot 1 \cdot 1 ~\backslash~ 7 | ||
| + | \\ | ||
| + | 8 ~/~ 1 \cdot 6 \cdot 1 \cdot 1 \cdot 1 \cdot 2 \cdot 1 ~\backslash~ 8 | ||
| + | \\ | ||
| + | 9 ~/~ 1 \cdot 7 \cdot 1 \cdot 25\cdot 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 9 | ||
| + | \\ | ||
| + | 10 ~/~ 1 \cdot 1 \cdot 1 \cdot 36\cdot 1 \cdot 2 \cdot 1 \cdot 8 \cdot 1 ~\backslash~ 10 | ||
| + | \end{matrix}</math> | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| + | |||
| + | <pre> | ||
| + | Example | ||
| + | |||
| + | * n o m | ||
| + | * \ / | ||
| + | * 1 . 1 | ||
| + | * \ / \ / | ||
| + | * 2 . 1 . 2 | ||
| + | * \ / \ / \ / | ||
| + | * 3 . 1 . 1 . 3 | ||
| + | * \ / \ / \ / \ / | ||
| + | * 4 . 1 . 2 . 1 . 4 | ||
| + | * \ / \ / \ / \ / \ / | ||
| + | * 5 . 1 . 3 . 1 . 1 . 5 | ||
| + | * \ / \ / \ / \ / \ / \ / | ||
| + | * 6 . 1 . 1 . 1 . 4 . 1 . 6 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 7 . 1 . 5 . 2 . 9 . 1 . 1 . 7 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10 | ||
| + | * | ||
| + | * Primal codes of finite partial functions on positive integers: | ||
| + | * 1 = { } | ||
| + | * 2 = 1:1 | ||
| + | * 3 = 2:1 | ||
| + | * 4 = 1:2 | ||
| + | * 5 = 3:1 | ||
| + | * 6 = 1:1 2:1 | ||
| + | * 7 = 4:1 | ||
| + | * 8 = 1:3 | ||
| + | * 9 = 2:2 | ||
| + | * 10 = 1:1 3:1 | ||
| + | * 11 = 5:1 | ||
| + | * 12 = 1:2 2:1 | ||
| + | * 13 = 6:1 | ||
| + | * 14 = 1:1 4:1 | ||
| + | * 15 = 2:1 3:1 | ||
| + | * 16 = 1:4 | ||
| + | * 17 = 7:1 | ||
| + | * 18 = 1:1 2:2 | ||
| + | * 19 = 8:1 | ||
| + | * 20 = 1:2 3:1 | ||
| + | </pre> | ||
| + | |||
| + | ==A106178== | ||
| + | |||
| + | * [http://oeis.org/wiki/A106178 A106178] | ||
| + | |||
| + | ===Wiki Table=== | ||
| + | |||
| + | {| align="center" style="font-weight:bold; text-align:center; width:90%" | ||
| + | | || || || || || || || || || || || || || || | ||
| + | | <font color="red">1</font> | ||
| + | | | ||
| + | | <font color="red">1</font> | ||
| + | |- | ||
| + | | || || || || || || || || || || || || || | ||
| + | | <font color="red">2</font> | ||
| + | | || · || | ||
| + | | <font color="red">2</font> | ||
| + | |- | ||
| + | | || || || || || || || || || || || || | ||
| + | | <font color="red">3</font> | ||
| + | | || · || || · || | ||
| + | | <font color="red">3</font> | ||
| + | |- | ||
| + | | || || || || || || || || || || || | ||
| + | | <font color="red">4</font> | ||
| + | | || · || || 2 || || · || | ||
| + | | <font color="red">4</font> | ||
| + | |- | ||
| + | | || || || || || || || || || || | ||
| + | | <font color="red">5</font> | ||
| + | | || · || || 3 || || 1 || || · || | ||
| + | | <font color="red">5</font> | ||
| + | |- | ||
| + | | || || || || || || || || || | ||
| + | | <font color="red">6</font> | ||
| + | | || · || || 1 || || 1 || || 4 || || · || | ||
| + | | <font color="red">6</font> | ||
| + | |- | ||
| + | | || || || || || || || || | ||
| + | | <font color="red">7</font> | ||
| + | | || · || || 5 || || 2 || || 9 || || 1 || || · || | ||
| + | | <font color="red">7</font> | ||
| + | |- | ||
| + | | || || || || || || || | ||
| + | | <font color="red">8</font> | ||
| + | | || · || || 6 || || 1 || || 1 || || 1 || || 2 || || · || | ||
| + | | <font color="red">8</font> | ||
| + | |- | ||
| + | | || || || || || || | ||
| + | | <font color="red">9</font> | ||
| + | | || · || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || · || | ||
| + | | <font color="red">9</font> | ||
| + | |- | ||
| + | | || || || || || | ||
| + | | <font color="red">10</font> | ||
| + | | || · || || 1 || || 1 || || 36|| || 1 || || 2 || || 1 || || 8 || || · || | ||
| + | | <font color="red">10</font> | ||
| + | |- | ||
| + | | || || || || | ||
| + | | <font color="red">11</font> | ||
| + | | || · || || 1 || || 1 || || 49 || || 1 || || 5 || || 1 || || 27 || || 1 || || · || | ||
| + | | <font color="red">11</font> | ||
| + | |- | ||
| + | | || || || | ||
| + | | <font color="red">12</font> | ||
| + | | || · || || 10 || || 3 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || · || | ||
| + | | <font color="red">12</font> | ||
| + | |- | ||
| + | | || || | ||
| + | | <font color="red">13</font> | ||
| + | | || · || || 11 || || 1 || || 1 || || 2 || || 7 || || 1 || || 125 || || 4 || || 3 || || 1 || || · || | ||
| + | | <font color="red">13</font> | ||
| + | |- | ||
| + | | || | ||
| + | | <font color="red">14</font> | ||
| + | | || · || || 3 || || 1 || || 100 || || 1 || || 1 || || 1 || || 216 || || 1 || || 1 || || 1 || || 4 || || · || | ||
| + | | <font color="red">14</font> | ||
| + | |- | ||
| + | | | ||
| + | | <font color="red">15</font> | ||
| + | | || · || || 13 || || 2 || || 121 || || 1 || || 3 || || 1 || || 343 || || 1 || || 5 || || 1 || || 9 || || 1 || || · || | ||
| + | | <font color="red">15</font> | ||
| + | |- | ||
| + | | width="3%" | <font color="red">16</font> | ||
| + | | width="3%" | | ||
| + | | width="3%" | · | ||
| + | | width="3%" | | ||
| + | | width="3%" | 14 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 9 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 10 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 6 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 2 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 2 | ||
| + | | width="3%" | | ||
| + | | width="3%" | · | ||
| + | | width="3%" | | ||
| + | | width="3%" | <font color="red">16</font> | ||
| + | |} | ||
| + | |||
| + | ===TeX Smallmatrix=== | ||
| + | |||
| + | {| align="center" | ||
| + | | | ||
| + | <math>\begin{smallmatrix} | ||
| + | &&&&&&&&&&&&&&& {\color{red}1} && {\color{red}1} | ||
| + | \\ | ||
| + | &&&&&&&&&&&&&& {\color{red}2} && \cdot & & {\color{red}2} | ||
| + | \\ | ||
| + | &&&&&&&&&&&&& {\color{red}3} && \cdot && \cdot && {\color{red}3} | ||
| + | \\ | ||
| + | &&&&&&&&&&&& {\color{red}4} && \cdot && 2 && \cdot && {\color{red}4} | ||
| + | \\ | ||
| + | &&&&&&&&&&& {\color{red}5} && \cdot && 3 && 1 && \cdot && {\color{red}5} | ||
| + | \\ | ||
| + | &&&&&&&&&& {\color{red}6} && \cdot && 1 && 1 && 4 && \cdot && {\color{red}6} | ||
| + | \\ | ||
| + | &&&&&&&&& {\color{red}7} && \cdot && 5 && 2 && 9 && 1 && \cdot && {\color{red}7} | ||
| + | \\ | ||
| + | &&&&&&&& {\color{red}8} && \cdot && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}8} | ||
| + | \\ | ||
| + | &&&&&&& {\color{red}9} && \cdot && 7 && 1 && 25 && 1 && 3 && 1 && \cdot && {\color{red}9} | ||
| + | \\ | ||
| + | &&&&&& {\color{red}10} && \cdot && 1 && 1 && 36 && 1 && 2 && 1 && 8 && \cdot && {\color{red}10} | ||
| + | \\ | ||
| + | &&&&& {\color{red}11} && \cdot && 1 && 1 && 49 && 1 && 5 && 1 && 27 && 1 && \cdot && {\color{red}11} | ||
| + | \\ | ||
| + | &&&& {\color{red}12} && \cdot && 10 && 3 && 1 && 1 && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}12} | ||
| + | \\ | ||
| + | &&& {\color{red}13} && \cdot && 11 && 1 && 1 && 2 && 7 && 1 && 125 && 4 && 3 && 1 && \cdot && {\color{red}13} | ||
| + | \\ | ||
| + | && {\color{red}14} && \cdot && 3 && 1 && 100 && 1 && 1 && 1 && 216 && 1 && 1 && 1 && 4 && \cdot && {\color{red}14} | ||
| + | \\ | ||
| + | & {\color{red}15} && \cdot && 13 && 2 && 121 && 1 && 3 && 1 && 343 && 1 && 5 && 1 && 9 && 1 && \cdot && {\color{red}15} | ||
| + | \\ | ||
| + | {\color{red}16} && \cdot && 14 && 1 && 9 && 1 && 10 && 1 && 1 && 1 && 6 && 1 && 2 && 1 && 2 && \cdot && {\color{red}16} | ||
| + | \end{smallmatrix}</math> | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| + | |||
| + | <pre> | ||
| + | Example | ||
| + | |||
| + | * n o m | ||
| + | * \ / | ||
| + | * 1 . 1 | ||
| + | * \ / \ / | ||
| + | * 2 . . 2 | ||
| + | * \ / \ / \ / | ||
| + | * 3 . . . 3 | ||
| + | * \ / \ / \ / \ / | ||
| + | * 4 . . 2 . . 4 | ||
| + | * \ / \ / \ / \ / \ / | ||
| + | * 5 . . 3 . 1 . . 5 | ||
| + | * \ / \ / \ / \ / \ / \ / | ||
| + | * 6 . . 1 . 1 . 4 . . 6 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 7 . . 5 . 2 . 9 . 1 . . 7 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 8 . . 6 . 1 . 1 . 1 . 2 . . 8 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 9 . . 7 . 1 . 25. 1 . 3 . 1 . . 9 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 10 . . 1 . 1 . 36. 1 . 2 . 1 . 8 . . 10 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 11 . . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 . . 11 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 12 . . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . . 12 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 13 . . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 . . 13 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 14 . . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 . . 14 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 15 . . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 . . 15 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 16 . . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 . . 16 | ||
| + | </pre> | ||
| + | |||
| + | ==A108352== | ||
| + | |||
| + | * [http://oeis.org/wiki/A108352 A108352] | ||
| + | |||
| + | ===Links=== | ||
| + | |||
| + | * Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002846.html Primal Code Characteristic, n = 1 to 1000] | ||
| + | * Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002847.html Primal Code Characteristic, n = 1001 to 2000] | ||
| + | * Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002853.html Primal Code Characteristic, n = 2001 to 3000] | ||
| + | |||
| + | ===TeX Array=== | ||
| + | |||
| + | {| align="center" | ||
| + | | | ||
| + | <math>\begin{array}{*{10}{l}} | ||
| + | a(1) | ||
| + | & = & 1 | ||
| + | & \text{because} & (\circ~ 1)^1 | ||
| + | & = & (\circ~ \varnothing)^1 | ||
| + | & = & 1. | ||
| + | \\ | ||
| + | a(2) | ||
| + | & = & 0 | ||
| + | & \text{because} & (\circ~ 2)^k | ||
| + | & = & (\circ~ 1\!:\!1)^k | ||
| + | & = & 2, | ||
| + | & \text{for all}~ k > 0. | ||
| + | \\ | ||
| + | a(3) | ||
| + | & = & 2 | ||
| + | & \text{because} & (\circ~ 3)^2 | ||
| + | & = & (\circ~ 2\!:\!1)^2 | ||
| + | & = & 1. | ||
| + | \\ | ||
| + | a(4) | ||
| + | & = & 2 | ||
| + | & \text{because} & (\circ~ 4 )^2 | ||
| + | & = & (\circ~ 1\!:\!2)^2 | ||
| + | & = &1. | ||
| + | \\ | ||
| + | a(5) | ||
| + | & = & 2 | ||
| + | & \text{because} & (\circ~ 5)^2 | ||
| + | & = & (\circ~ 3\!:\!1)^2 | ||
| + | & = & 1. | ||
| + | \\ | ||
| + | a(6) | ||
| + | & = & 0 | ||
| + | & \text{because} & (\circ~ 6)^k | ||
| + | & = & (\circ~ 1\!:\!1 ~~ 2\!:\!1)^k | ||
| + | & = & 6, | ||
| + | & \text{for all}~ k > 0. | ||
| + | \\ | ||
| + | a(7) | ||
| + | & = & 2 | ||
| + | & \text{because} & (\circ~ 7)^2 | ||
| + | & = & (\circ~ 4\!:\!1)^1 | ||
| + | & = & 1. | ||
| + | \\ | ||
| + | a(8) | ||
| + | & = & 2 | ||
| + | & \text{because} & (\circ~ 8)^2 | ||
| + | & = & (\circ~ 1\!:\!3)^1 | ||
| + | & = & 1. | ||
| + | \\ | ||
| + | a(9) | ||
| + | & = & 0 | ||
| + | & \text{because} & (\circ~ 9)^k | ||
| + | & = & (\circ~ 2\!:\!2)^k | ||
| + | & = & 9, | ||
| + | & \text{for all}~ k > 0. | ||
| + | \\ | ||
| + | a(10) | ||
| + | & = & 0 | ||
| + | & \text{because} & (\circ~ 10)^k | ||
| + | & = & (\circ~ 1\!:\!1 ~~ 3\!:\!1)^k | ||
| + | & = & 10, | ||
| + | & \text{for all}~ k > 0. | ||
| + | \end{array}</math> | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| + | |||
| + | <pre> | ||
| + | Example | ||
| + | |||
| + | * a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1. | ||
| + | * a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k. | ||
| + | * a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1. | ||
| + | * a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1. | ||
| + | * a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1. | ||
| + | * a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k. | ||
| + | * a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1. | ||
| + | * a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1. | ||
| + | * a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k. | ||
| + | * a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k. | ||
| + | * Detail of calculation for compositional powers of 12: | ||
| + | * (12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18 | ||
| + | * (12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12 | ||
| + | * Detail of calculation for compositional powers of 20: | ||
| + | * (20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25 | ||
| + | * (20 o)^3 = (3:2) o (1:2 3:1) = 1 | ||
| + | </pre> | ||
| + | |||
| + | ==A108371== | ||
| + | |||
| + | * [http://oeis.org/wiki/A108371 A108371] | ||
| + | |||
| + | ===Wiki Table=== | ||
| + | |||
| + | {| align="center" style="font-weight:bold; text-align:center; width:90%" | ||
| + | | || || || || || || || || || || || || || || | ||
| + | | <font color="red">1</font> | ||
| + | | | ||
| + | | <font color="red">1</font> | ||
| + | |- | ||
| + | | || || || || || || || || || || || || || | ||
| + | | <font color="red">2</font> | ||
| + | | || 1 || | ||
| + | | <font color="red">2</font> | ||
| + | |- | ||
| + | | || || || || || || || || || || || || | ||
| + | | <font color="red">3</font> | ||
| + | | || 2 || || 1 || | ||
| + | | <font color="red">3</font> | ||
| + | |- | ||
| + | | || || || || || || || || || || || | ||
| + | | <font color="red">4</font> | ||
| + | | || 3 || || 2 || || 1 || | ||
| + | | <font color="red">4</font> | ||
| + | |- | ||
| + | | || || || || || || || || || || | ||
| + | | <font color="red">5</font> | ||
| + | | || 4 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">5</font> | ||
| + | |- | ||
| + | | || || || || || || || || || | ||
| + | | <font color="red">6</font> | ||
| + | | || 5 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">6</font> | ||
| + | |- | ||
| + | | || || || || || || || || | ||
| + | | <font color="red">7</font> | ||
| + | | || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">7</font> | ||
| + | |- | ||
| + | | || || || || || || || | ||
| + | | <font color="red">8</font> | ||
| + | | || 7 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">8</font> | ||
| + | |- | ||
| + | | || || || || || || | ||
| + | | <font color="red">9</font> | ||
| + | | || 8 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">9</font> | ||
| + | |- | ||
| + | | || || || || || | ||
| + | | <font color="red">10</font> | ||
| + | | || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">10</font> | ||
| + | |- | ||
| + | | || || || || | ||
| + | | <font color="red">11</font> | ||
| + | | || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">11</font> | ||
| + | |- | ||
| + | | || || || | ||
| + | | <font color="red">12</font> | ||
| + | | || 11|| || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">12</font> | ||
| + | |- | ||
| + | | || || | ||
| + | | <font color="red">13</font> | ||
| + | | || 12|| || 1 || || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">13</font> | ||
| + | |- | ||
| + | | || | ||
| + | | <font color="red">14</font> | ||
| + | | || 13|| || 18|| || 1 || || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">14</font> | ||
| + | |- | ||
| + | | | ||
| + | | <font color="red">15</font> | ||
| + | | || 14 || || 1 || || 12 || || 1 || || 10 || || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | ||
| + | | <font color="red">15</font> | ||
| + | |- | ||
| + | | width="3%" | <font color="red">16</font> | ||
| + | | width="3%" | | ||
| + | | width="3%" | 15 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 14 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 18 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 10 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 9 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 6 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 2 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | <font color="red">16</font> | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| + | |||
| + | <pre> | ||
| + | Example | ||
| + | |||
| + | * Table: T(n,k) = (n o)^k | ||
| + | * T(n,k) | ||
| + | * \ / | ||
| + | * 1 . 1 | ||
| + | * \ / \ / | ||
| + | * 2 . 1 . 2 | ||
| + | * \ / \ / \ / | ||
| + | * 3 . 2 . 1 . 3 | ||
| + | * \ / \ / \ / \ / | ||
| + | * 4 . 3 . 2 . 1 . 4 | ||
| + | * \ / \ / \ / \ / \ / | ||
| + | * 5 . 4 . 1 . 2 . 1 . 5 | ||
| + | * \ / \ / \ / \ / \ / \ / | ||
| + | * 6 . 5 . 1 . 1 . 2 . 1 . 6 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 7 . 6 . 1 . 1 . 1 . 2 . 1 . 7 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 8 . 7 . 6 . 1 . 1 . 1 . 2 . 1 . 8 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 9 . 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 9 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 10 . 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 10 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 11 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 11 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 12 . 11. 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 12 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 13 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 13 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 14 . 13. 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 14 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 15 . 14. 1 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 15 | ||
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | ||
| + | * 16 . 15. 14. 1 . 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 16 | ||
| + | </pre> | ||
| + | |||
| + | ==A109300== | ||
| + | |||
| + | * [http://oeis.org/wiki/A109300 A109300] | ||
| + | |||
| + | ===JPEG=== | ||
| + | |||
| + | {| align="center" border="1" cellpadding="10" | ||
| + | | | ||
| + | <p>[[Image:Rote 3 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p> | ||
| + | | | ||
| + | <p>[[Image:Rote 4 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p> | ||
| + | | | ||
| + | <p>[[Image:Rote 6 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p> | ||
| + | | | ||
| + | <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p> | ||
| + | | | ||
| + | <p>[[Image:Rote 12 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p> | ||
| + | | | ||
| + | <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p> | ||
| + | | | ||
| + | <p>[[Image:Rote 36 Big.jpg|145px]]</p><br> | ||
| + | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p> | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| + | |||
| + | <pre> | ||
| + | Example | ||
| + | |||
| + | * Table of Rotes and Primal Functions for Positive Integers of Rote Height 2 | ||
| + | * | ||
| + | * o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | | | | | | | ||
| + | * o-o o-o o-o o-o o---o o-o o-o o-o o---o o-o o---o | ||
| + | * | | | | | | | | | | | | ||
| + | * O O O===O O O=====O O===O O=====O | ||
| + | * | ||
| + | * 2:1 1:2 1:1 2:1 2:2 1:2 2:1 1:1 2:2 1:2 2:2 | ||
| + | * | ||
| + | * 3 4 6 9 12 18 36 | ||
| + | * | ||
| + | </pre> | ||
| + | |||
| + | ==A109301== | ||
| + | |||
| + | * [http://oeis.org/wiki/A109301 A109301] | ||
| + | |||
| + | ===Example=== | ||
| + | |||
| + | : <math>802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1</math> | ||
| + | |||
| + | : <math>\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}</math> | ||
| + | |||
| + | : <math>\begin{array}{lllll} | ||
| + | 802701 | ||
| + | & = & 9 \cdot 89189 | ||
| + | & = & 2\!:\!2 ~~ 8638\!:\!1 | ||
| + | \\ | ||
| + | 8638 | ||
| + | & = & 2 \cdot 7 \cdot 617 | ||
| + | & = & 1\!:\!1 ~~ 4\!:\!1 ~~ 113\!:\!1 | ||
| + | \\ | ||
| + | 113 | ||
| + | & & | ||
| + | & = & 30\!:\!1 | ||
| + | \\ | ||
| + | 30 | ||
| + | & = & 2 \cdot 3 \cdot 5 | ||
| + | & = & 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 | ||
| + | \\ | ||
| + | 4 | ||
| + | & & | ||
| + | & = & 1\!:\!2 | ||
| + | \\ | ||
| + | 3 | ||
| + | & & | ||
| + | & = & 2\!:\!1 | ||
| + | \\ | ||
| + | 2 | ||
| + | & & | ||
| + | & = & 1\!:\!1 | ||
| + | \end{array}</math> | ||
| + | |||
| + | : <math>\text{So the rote of 802701 is the following graph:}\!</math> | ||
| + | |||
| + | :{| border="1" cellpadding="20" | ||
| + | | [[Image:Rote 802701 Big.jpg|330px]] | ||
| + | |} | ||
| + | |||
| + | : <math>\text{By inspection, the rote height of 802701 is 6.}\!</math> | ||
| + | |||
| + | ===JPEG=== | ||
| + | |||
| + | {| align="center" border="1" cellpadding="6" | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 1 Big.jpg|20px]]</p><br> | ||
| + | <p><math>1\!</math></p><br> | ||
| + | <p><math>a(1) ~=~ 0</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 2 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}\!</math></p><br> | ||
| + | <p><math>a(2) ~=~ 1</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 3 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(3) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 4 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(4) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 5 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(5) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 6 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(6) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 7 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(7) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 8 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(8) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(9) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 10 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(10) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 11 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(11) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 12 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(12) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 13 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(13) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 14 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(14) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 15 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(15) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 16 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(16) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 17 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(17) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(18) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 19 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(19) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 20 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(20) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 21 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(21) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 22 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(22) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 23 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(23) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 24 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(24) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 25 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(25) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 26 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(26) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 27 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(27) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 28 Big.jpg|130px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(28) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 29 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(29) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 30 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(30) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 31 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br> | ||
| + | <p><math>a(31) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 32 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(32) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 33 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(33) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 34 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(34) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 35 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(35) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 36 Big.jpg|145px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(36) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 37 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(37) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 38 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(38) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 39 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(39) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 40 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(40) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 41 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(41) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 42 Big.jpg|145px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(42) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 43 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(43) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 44 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(44) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 45 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(45) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 46 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(46) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 47 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(47) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 48 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(48) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 49 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(49) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 50 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> | ||
| + | <p><math>a(50) ~=~ 3</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 51 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(51) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 52 Big.jpg|145px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(52) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 53 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(53) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 54 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(54) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 55 Big.jpg|80px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(55) ~=~ 4</math></p> | ||
| + | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 56 Big.jpg|130px]]</p><br> | ||
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(56) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 57 Big.jpg|105px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(57) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 58 Big.jpg|120px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(58) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 59 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br> | ||
| + | <p><math>a(59) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Rote 60 Big.jpg|155px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(60) ~=~ 3</math></p> | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| + | |||
| + | <pre> | ||
| + | Comment | ||
| + | |||
| + | * Table of Rotes and Primal Functions for Positive Integers from 1 to 40 | ||
* | * | ||
| − | * 36 37 38 39 40 | + | * o-o |
| − | * | + | * | |
| − | * In these Figures, "extended lines of identity" like o===o | + | * o-o o-o o-o |
| − | * indicate identified nodes and capital O is the root node. | + | * | | | |
| − | * The rote height in gammas is found by finding the number | + | * o-o o-o o-o o-o |
| − | * of graphs of the following shape between the root and one | + | * | | | | |
| − | * of the highest nodes of the tree: | + | * O O O O O |
| − | * o--o | + | * |
| − | * | | + | * { } 1:1 2:1 1:2 3:1 |
| − | * o | + | * |
| − | * A sequence like this, that can be regarded as a nonnegative integer | + | * 1 2 3 4 5 |
| − | * measure on positive integers, may have as many as 3 other sequences | + | * |
| − | * associated with it. Given that the fiber of a function f at n is all | + | * |
| − | * the domain elements that map to n, we always have the fiber minimum | + | * o-o o-o o-o |
| − | * or minimum inverse function and may also have the fiber cardinality | + | * | | | |
| − | * and the fiber maximum or maximum inverse function. For A109301, the | + | * o-o o-o o-o o-o o-o o-o |
| − | * minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the | + | * | | | | | | |
| − | * first positive integer whose rote height is n, the fiber cardinality | + | * o-o o-o o-o o-o o---o o-o o-o |
| − | * is A109300, giving the number of positive integers of rote height n, | + | * | | | | | | | |
| − | * while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving | + | * O===O O O O O===O |
| − | * the last positive integer whose rote height is n, has the following | + | * |
| − | * initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36, | + | * 1:1 2:1 4:1 1:3 2:2 1:1 3:1 |
| − | * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 = | + | * |
| − | * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly | + | * 6 7 8 9 10 |
| − | * 7.840858554516122655953405327738 x 10^371. | + | * |
| + | * | ||
| + | * o-o | ||
| + | * | | ||
| + | * o-o o-o o-o o-o | ||
| + | * | | | | | ||
| + | * o-o o-o o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | | | ||
| + | * o-o o-o o-o o===o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | | | ||
| + | * O O=====O O O===O O===O | ||
| + | * | ||
| + | * 5:1 1:2 2:1 6:1 1:1 4:1 2:1 3:1 | ||
| + | * | ||
| + | * 11 12 13 14 15 | ||
| + | * | ||
| + | * | ||
| + | * o-o o-o | ||
| + | * | | | ||
| + | * o-o o-o o-o o-o | ||
| + | * | | | | | ||
| + | * o-o o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | | ||
| + | * o-o o-o o-o o---o o-o o-o o-o | ||
| + | * | | | | | | | | ||
| + | * O O O===O O O=====O | ||
| + | * | ||
| + | * 1:4 7:1 1:1 2:2 8:1 1:2 3:1 | ||
| + | * | ||
| + | * 16 17 18 19 20 | ||
| + | * | ||
| + | * | ||
| + | * o-o | ||
| + | * | | ||
| + | * o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | ||
| + | * o-o o-o o-o o---o o-o o-o o-o o-o | ||
| + | * | | | | | | | | | ||
| + | * o-o o-o o-o o-o o-o o-o o-o o---o | ||
| + | * | | | | | | | | | ||
| + | * O===O O===O O O=====O O | ||
| + | * | ||
| + | * 2:1 4:1 1:1 5:1 9:1 1:3 2:1 3:2 | ||
| + | * | ||
| + | * 21 22 23 24 25 | ||
| + | * | ||
| + | * | ||
| + | * o-o | ||
| + | * | | ||
| + | * o-o o-o o-o o-o o-o | ||
| + | * | | | | | | ||
| + | * o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | | | | | ||
| + | * o-o o===o-o o---o o-o o-o o===o-o o-o o-o o-o | ||
| + | * | | | | | | | | | | ||
| + | * O===O O O=====O O O===O===O | ||
| + | * | ||
| + | * 1:1 6:1 2:3 1:2 4:1 10:1 1:1 2:1 3:1 | ||
| + | * | ||
| + | * 26 27 28 29 30 | ||
| + | * | ||
| + | * | ||
| + | * o-o | ||
| + | * | | ||
| + | * o-o o-o o-o o-o | ||
| + | * | | | | | ||
| + | * o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | ||
| + | * o-o o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | | ||
| + | * o-o o-o o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | | | ||
| + | * O O O===O O===O O===O | ||
| + | * | ||
| + | * 11:1 1:5 2:1 5:1 1:1 7:1 3:1 4:1 | ||
| + | * | ||
| + | * 31 32 33 34 35 | ||
| + | * | ||
| + | * | ||
| + | * o-o | ||
| + | * | | ||
| + | * o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | ||
| + | * o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o | ||
| + | * | | | | | | | | | | | | ||
| + | * o-o o---o o=====o-o o-o o-o o-o o===o-o o-o o-o | ||
| + | * | | | | | | | | | | ||
| + | * O=====O O O===O O===O O=====O | ||
| + | * | ||
| + | * 1:2 2:2 12:1 1:1 8:1 2:1 6:1 1:3 3:1 | ||
| + | * | ||
| + | * 36 37 38 39 40 | ||
| + | * | ||
| + | * In these Figures, "extended lines of identity" like o===o | ||
| + | * indicate identified nodes and capital O is the root node. | ||
| + | * The rote height in gammas is found by finding the number | ||
| + | * of graphs of the following shape between the root and one | ||
| + | * of the highest nodes of the tree: | ||
| + | * o--o | ||
| + | * | | ||
| + | * o | ||
| + | * A sequence like this, that can be regarded as a nonnegative integer | ||
| + | * measure on positive integers, may have as many as 3 other sequences | ||
| + | * associated with it. Given that the fiber of a function f at n is all | ||
| + | * the domain elements that map to n, we always have the fiber minimum | ||
| + | * or minimum inverse function and may also have the fiber cardinality | ||
| + | * and the fiber maximum or maximum inverse function. For A109301, the | ||
| + | * minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the | ||
| + | * first positive integer whose rote height is n, the fiber cardinality | ||
| + | * is A109300, giving the number of positive integers of rote height n, | ||
| + | * while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving | ||
| + | * the last positive integer whose rote height is n, has the following | ||
| + | * initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36, | ||
| + | * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 = | ||
| + | * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly | ||
| + | * 7.840858554516122655953405327738 x 10^371. | ||
| + | |||
| + | Example | ||
| + | |||
| + | * Writing (prime(i))^j as i:j, we have: | ||
| + | * 802701 = 2:2 8638:1 | ||
| + | * 8638 = 1:1 4:1 113:1 | ||
| + | * 113 = 30:1 | ||
| + | * 30 = 1:1 2:1 3:1 | ||
| + | * 4 = 1:2 | ||
| + | * 3 = 2:1 | ||
| + | * 2 = 1:1 | ||
| + | * 1 = { } | ||
| + | * So rote(802701) is the graph: | ||
| + | * | ||
| + | * o-o | ||
| + | * | | ||
| + | * o-o o-o | ||
| + | * | | | ||
| + | * o-o o-o o-o o-o | ||
| + | * | | | | | ||
| + | * o-o o===o===o-o | ||
| + | * | | | ||
| + | * o-o o-o o-o o-o o---------o | ||
| + | * | | | | | | ||
| + | * o---o o===o=====o---------o | ||
| + | * | | | ||
| + | * O=======O | ||
| + | * | ||
| + | * Therefore rhig(802701) = 6. | ||
</pre> | </pre> | ||
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===JPEG=== | ===JPEG=== | ||
| − | |||
| − | |||
{| align="center" border="1" cellpadding="10" | {| align="center" border="1" cellpadding="10" | ||
| Line 1,225: | Line 3,038: | ||
<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p> | <p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p> | ||
|} | |} | ||
| − | |||
| − | |||
===ASCII=== | ===ASCII=== | ||
Latest revision as of 18:48, 31 January 2010
A061396
Plain Wiki Table
Large Scale
| \(\text{Integer}\!\) | \(\text{Factorization}\!\) | \(\text{Notation}\!\) | \(\text{Riff Digraph}\!\) | \(\text{Rote Graph}\!\) | \(\text{Traversal}\!\) |
| \(1\!\) | \(1\!\) | 
|
|||
| \(2\!\) | \(\text{p}_1^1\!\) | \(\text{p}\!\) | 
|
|
\(((~))\) |
| \(3\!\) |
\(\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}\) |
\(\text{p}_\text{p}\!\) | 
|
|
\((((~))(~))\) |
| \(4\!\) |
\(\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}\) |
\(\text{p}^\text{p}\!\) | 
|
|
\(((((~))))\) |
| \(5\!\) |
\(\begin{array}{lll}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\'"`UNIQ-MathJax1-QINU`"'
'"`UNIQ-MathJax2-QINU`"'
'"`UNIQ-MathJax3-QINU`"'
'"`UNIQ-MathJax4-QINU`"'
:{| border="1" cellpadding="20"
| [[Image:Rote 802701 Big.jpg|330px]]
|}
'"`UNIQ-MathJax5-QINU`"'
==='"`UNIQ--h-40--QINU`"'JPEG===
{| align="center" border="1" cellpadding="6"
| valign="bottom" |
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
<p>\(1\!\) \(a(1) ~=~ 0\) |
\(\text{p}\!\) \(a(2) ~=~ 1\) |
\(\text{p}_\text{p}\!\) \(a(3) ~=~ 2\) |
\(\text{p}^\text{p}\!\) \(a(4) ~=~ 2\) |
\(\text{p}_{\text{p}_\text{p}}\!\) \(a(5) ~=~ 3\) |
|
\(\text{p} \text{p}_\text{p}\!\) \(a(6) ~=~ 2\) |
\(\text{p}_{\text{p}^\text{p}}\!\) \(a(7) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}}\!\) \(a(8) ~=~ 3\) |
\(\text{p}_\text{p}^\text{p}\!\) \(a(9) ~=~ 2\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(10) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(11) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(a(12) ~=~ 2\) |
\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(13) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(14) ~=~ 3\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(15) ~=~ 3\) | |
|
\(\text{p}^{\text{p}^\text{p}}\!\) \(a(16) ~=~ 3\) |
\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(17) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(18) ~=~ 2\) |
\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(19) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(20) ~=~ 3\) | |
|
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(21) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(22) ~=~ 4\) |
\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(23) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(a(24) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(25) ~=~ 3\) | |
|
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(26) ~=~ 3\) |
\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(27) ~=~ 3\) |
\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(28) ~=~ 3\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(29) ~=~ 4\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(30) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(a(31) ~=~ 5\) |
\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(a(32) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(33) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(34) ~=~ 4\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(35) ~=~ 3\) | |
|
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(36) ~=~ 2\) |
\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(a(37) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(38) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(39) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(a(40) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(a(41) ~=~ 4\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(42) ~=~ 3\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(a(43) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(44) ~=~ 4\) |
\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(45) ~=~ 3\) | |
|
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(46) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(47) ~=~ 4\) |
\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(a(48) ~=~ 3\) |
\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(a(49) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(50) ~=~ 3\) | |
|
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(51) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(52) ~=~ 3\) |
\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(a(53) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(54) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(55) ~=~ 4\) | |
|
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(56) ~=~ 3\) |
\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(57) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(58) ~=~ 4\) |
\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(a(59) ~=~ 5\) |
\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(60) ~=~ 3\) |
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
|
\(\begin{array}{l} \varnothing \\ 1 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\) |
\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\) |
|
\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\) |
\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\) |
\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\) |
\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\) |
\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\) |
|
\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\) |
\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\) |
\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\) |
\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\) |
\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\) |
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
\(\text{Writing}~ \operatorname{prime}(i)^j ~\text{as}~ i\!:\!j, 2500 = 4 \cdot 625 = 2^2 5^4 = 1\!:\!2 ~~ 3\!:\!4 ~\text{has the following rote:}\)
|
\(\text{So}~ a(2500) = a(1\!:\!2 ~~ 3\!:\!4) = a(1) + a(2) + a(3) + a(4) + 1 = 1 + 3 + 5 + 5 + 1 = 15.\)
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.
