Difference between revisions of "User:Jon Awbrey/SEQUENCES"
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{| 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> | ||
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{| 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> | ||
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{| align="center" border="1" width="96%" | {| align="center" border="1" width="96%" | ||
− | |+ style="height: | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
|- style="height:50px; background:#f0f0ff" | |- style="height:50px; background:#f0f0ff" | ||
| | | | ||
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{| align="center" border="1" width="96%" | {| align="center" border="1" width="96%" | ||
− | |+ style="height: | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
|- style="height:50px; background:#f0f0ff" | |- style="height:50px; background:#f0f0ff" | ||
| | | | ||
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\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math> | | <math>\text{p}_{\text{p}^{\text{p}}}\!</math> | ||
− | | | + | | [[Image:Riff 7 Big.jpg|38px]] |
| [[Image:Rote 7 Big.jpg|38px]] | | [[Image:Rote 7 Big.jpg|38px]] | ||
| <math>(((((~))))(~))</math> | | <math>(((((~))))(~))</math> | ||
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\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math> | | <math>\text{p}^{\text{p}_{\text{p}}}\!</math> | ||
− | | | + | | [[Image:Riff 8 Big.jpg|38px]] |
| [[Image:Rote 8 Big.jpg|38px]] | | [[Image:Rote 8 Big.jpg|38px]] | ||
| <math>(((((~))(~))))</math> | | <math>(((((~))(~))))</math> | ||
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\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}_\text{p}^\text{p}\!</math> | | <math>\text{p}_\text{p}^\text{p}\!</math> | ||
− | | | + | | [[Image:Riff 9 Big.jpg|24px]] |
| [[Image:Rote 9 Big.jpg|48px]] | | [[Image:Rote 9 Big.jpg|48px]] | ||
| <math>(((~))(((~))))</math> | | <math>(((~))(((~))))</math> | ||
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\end{array}</math> | \end{array}</math> | ||
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math> | | <math>\text{p}^{\text{p}^{\text{p}}}\!</math> | ||
− | | | + | | [[Image:Riff 16 Big.jpg|38px]] |
| [[Image:Rote 16 Big.jpg|52px]] | | [[Image:Rote 16 Big.jpg|52px]] | ||
| <math>((((((~))))))</math> | | <math>((((((~))))))</math> | ||
Line 1,015: | Line 1,015: | ||
</pre> | </pre> | ||
− | == | + | ==A062504== |
+ | |||
+ | * [http://oeis.org/wiki/A062504 A062504] | ||
+ | |||
+ | ===TeX Array=== | ||
− | + | {| align="center" | |
+ | | | ||
+ | <math>\begin{array}{l|l|r} | ||
+ | k | ||
+ | & P_k | ||
+ | = \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \} | ||
+ | = \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \} | ||
+ | & |P_k| | ||
+ | \\[10pt] | ||
+ | 0 & \{ 1 \} & 1 | ||
+ | \\ | ||
+ | 1 & \{ 2 \} & 1 | ||
+ | \\ | ||
+ | 2 & \{ 3, 4 \} & 2 | ||
+ | \\ | ||
+ | 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 | ||
+ | \\ | ||
+ | 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 | ||
+ | \end{array}</math> | ||
+ | |} | ||
===JPEG=== | ===JPEG=== | ||
− | + | {| align="center" border="1" width="90%" | |
− | + | |+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math> | |
− | {| align="center" border="1" | + | |- 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]] | |
− | <p><math> | + | |- |
− | + | | <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 | |
− | ===ASCII=== | + | \\[12pt] |
− | + | & = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1 | |
− | <pre> | + | \end{array}</math> |
− | Comment | + | | <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math> |
− | + | | [[Image:Riff 53 Big.jpg|90px]] | |
− | * Table of Rotes and Primal Functions for Positive Integers from 1 to 40 | + | | [[Image:Rote 53 Big.jpg|90px]] |
− | * | + | |- |
− | * o-o | + | | <math>64\!</math> |
− | * | | + | | |
− | * o-o o-o o-o | + | <math>\begin{array}{lll} |
− | * | | | | + | \text{p}_1^6 |
− | * o-o o-o o-o o-o | + | & = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1} |
− | * | | | | | + | \\[12pt] |
− | * O O O O O | + | & = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1} |
− | * | + | \end{array}</math> |
− | * { } 1:1 2:1 1:2 3:1 | + | | <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math> |
− | * | + | | [[Image:Riff 64 Big.jpg|65px]] |
− | * 1 2 3 4 5 | + | | [[Image:Rote 64 Big.jpg|105px]] |
− | * | + | |- |
− | * | + | | <math>81\!</math> |
− | * o-o o-o o-o | + | | |
− | * | | | | + | <math>\begin{array}{lll} |
− | * o-o o-o o-o o-o o-o o-o | + | \text{p}_2^4 |
− | * | | | | | | | + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2} |
− | * o-o o-o o-o o-o o---o o-o o-o | + | \\[12pt] |
− | * | | | | | | | | + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}} |
− | * O===O O O O O===O | + | \end{array}</math> |
− | * | + | | <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math> |
− | * 1:1 2:1 4:1 1:3 2:2 1:1 3:1 | + | | [[Image:Riff 81 Big.jpg|65px]] |
− | * | + | | [[Image:Rote 81 Big.jpg|105px]] |
− | * 6 7 8 9 10 | + | |- |
− | * | + | | <math>128\!</math> |
− | * | + | | |
− | * o-o | + | <math>\begin{array}{lll} |
− | * | | + | \text{p}_1^7 |
− | * o-o o-o o-o o-o | + | & = & \text{p}_1^{\text{p}_4^1} |
− | * | | | | | + | \\[12pt] |
− | * o-o o-o o-o o-o o-o o-o o-o o-o | + | & = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1} |
− | * | | | | | | | | | + | \\[12pt] |
− | * o-o o-o o-o o===o-o o-o o-o o-o o-o | + | & = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1} |
− | * | | | | | | | | | + | \end{array}</math> |
− | * O O=====O O O===O O===O | + | | <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math> |
− | * | + | | [[Image:Riff 128 Big.jpg|90px]] |
− | * 5:1 1:2 2:1 6:1 1:1 4:1 2:1 3:1 | + | | [[Image:Rote 128 Big.jpg|90px]] |
− | * | + | |- |
− | * 11 12 13 14 15 | + | | <math>256\!</math> |
− | * | + | | |
− | * | + | <math>\begin{array}{lll} |
− | * o-o o-o | + | \text{p}_1^8 |
− | * | | | + | & = & \text{p}_1^{\text{p}_1^3} |
− | * o-o o-o o-o o-o | + | \\[12pt] |
− | * | | | | | + | & = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}} |
− | * o-o o-o o-o o-o o-o o-o o-o | + | \\[12pt] |
− | * | | | | | | | | + | & = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}} |
− | * o-o o-o o-o o---o o-o o-o o-o | + | \end{array}</math> |
− | * | | | | | | | | + | | <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math> |
− | * O O O===O O O=====O | + | | [[Image:Riff 256 Big.jpg|90px]] |
− | * | + | | [[Image:Rote 256 Big.jpg|90px]] |
− | * 1:4 7:1 1:1 2:2 8:1 1:2 3:1 | + | |- |
− | * | + | | <math>512\!</math> |
− | * 16 17 18 19 20 | + | | |
− | * | + | <math>\begin{array}{lll} |
− | * | + | \text{p}_1^9 |
− | * o-o | + | & = & \text{p}_1^{\text{p}_2^2} |
− | * | | + | \\[12pt] |
− | * o-o o-o o-o o-o o-o o-o | + | & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}} |
− | * | | | | | | | + | \end{array}</math> |
− | * o-o o-o o-o o---o o-o o-o o-o o-o | + | | <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math> |
− | * | | | | | | | | | + | | [[Image:Riff 512 Big.jpg|65px]] |
− | * o-o o-o o-o o-o o-o o-o o-o o---o | + | | [[Image:Rote 512 Big.jpg|105px]] |
− | * | | | | | | | | | + | |- |
− | * O===O O===O O O=====O O | + | | <math>65536\!</math> |
− | * | + | | |
− | * 2:1 4:1 1:1 5:1 9:1 1:3 2:1 3:2 | + | <math>\begin{array}{lll} |
− | * | + | \text{p}_1^{16} |
− | * 21 22 23 24 25 | + | & = & \text{p}_1^{\text{p}_1^4} |
− | * | + | \\[12pt] |
− | * | + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}} |
− | * o-o | + | \\[12pt] |
− | * | | + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}} |
− | * o-o o-o o-o o-o o-o | + | \end{array}</math> |
− | * | | | | | | + | | <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math> |
− | * o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o | + | | [[Image:Riff 65536 Big.jpg|90px]] |
− | * | | | | | | | | | | | + | | [[Image:Rote 65536 Big.jpg|115px]] |
− | * o-o o===o-o o---o o-o o-o o===o-o o-o o-o o-o | + | |} |
− | * | | | | | | | | | | + | |} |
− | * O===O O O=====O O O===O===O | + | |
− | * | + | ===ASCII=== |
− | * 1:1 6:1 2:3 1:2 4:1 10:1 1:1 2:1 3:1 | + | |
− | * | + | <pre> |
− | * 26 27 28 29 30 | + | Example |
− | * | + | |
− | * | + | * k | natural numbers n such that |riff(n)| = k |
− | * o-o | + | * 0 | 1; |
− | * | | + | * 1 | 2; |
− | * o-o o-o o-o o-o | + | * 2 | 3, 4; |
− | * | | | | | + | * 3 | 5, 6, 7, 8, 9, 16; |
− | * o-o o-o o-o o-o o-o o-o | + | * 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 o-o o-o o-o o-o o-o | + | * .................o.......o..o.......o |
− | * | | | | | | | | + | * .................|.......^..|.......^ |
− | * o-o o-o o-o o-o o-o o-o o-o o-o | + | * .................v.......|..v.......| |
− | * | | | | | | | | | + | * ...........o..o..o....o..o..o..o.o..o |
− | * O O O===O O===O O===O | + | * ...........|..^..|....|..|..^..|.^..^ |
− | * | + | * ...........v..|..v....v..v..|..v/...| |
− | * 11:1 1:5 2:1 5:1 1:1 7:1 3:1 4:1 | + | * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x; |
− | * | + | * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16; |
− | * 31 32 33 34 35 | + | </pre> |
− | * | + | |
− | * | + | ==A062537== |
− | * o-o | + | |
− | * | | + | * [http://oeis.org/wiki/A062537 A062537] |
− | * o-o o-o o-o o-o o-o o-o | + | |
− | * | | | | | | | + | ===Wiki + TeX + JPEG=== |
− | * o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o | + | |
− | * | | | | | | | | | | | | + | {| align="center" border="1" cellpadding="10" |
− | * o-o o---o o=====o-o o-o o-o o-o o===o-o o-o o-o | + | |+ style="height:25px" | <math>a(n) = \text{Number of Nodes in the Riff of}~ n</math> |
− | * | | | | | | | | | | + | | valign="bottom" | |
− | * O=====O O O===O O===O O=====O | + | <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 | ||
+ | * | ||
+ | * 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 | * 1:2 2:2 12:1 1:1 8:1 2:1 6:1 1:3 3:1 | ||
− | * | + | * |
− | * 36 37 38 39 40 | + | * 36 37 38 39 40 |
− | * | + | * |
− | * In these Figures, "extended lines of identity" like o===o | + | * In these Figures, "extended lines of identity" like o===o |
− | * indicate identified nodes and capital O is the root node. | + | * indicate identified nodes and capital O is the root node. |
− | * The rote height in gammas is found by finding the number | + | * The rote height in gammas is found by finding the number |
− | * of graphs of the following shape between the root and one | + | * of graphs of the following shape between the root and one |
− | * of the highest nodes of the tree: | + | * of the highest nodes of the tree: |
− | * o--o | + | * o--o |
− | * | | + | * | |
− | * o | + | * o |
− | * A sequence like this, that can be regarded as a nonnegative integer | + | * A sequence like this, that can be regarded as a nonnegative integer |
− | * measure on positive integers, may have as many as 3 other sequences | + | * 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 | + | * 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 | + | * the domain elements that map to n, we always have the fiber minimum |
− | * or minimum inverse function and may also have the fiber cardinality | + | * or minimum inverse function and may also have the fiber cardinality |
− | * and the fiber maximum or maximum inverse function. For A109301, the | + | * and the fiber maximum or maximum inverse function. For A109301, the |
− | * minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the | + | * minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the |
− | * first positive integer whose rote height is n, the fiber cardinality | + | * first positive integer whose rote height is n, the fiber cardinality |
− | * is A109300, giving the number of positive integers of rote height n, | + | * is A109300, giving the number of positive integers of rote height n, |
− | * while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving | + | * while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving |
− | * the last positive integer whose rote height is n, has the following | + | * 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, | + | * 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 = | + | * 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 | + | * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly |
− | * 7.840858554516122655953405327738 x 10^371. | + | * 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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<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p> | <p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p> | ||
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===ASCII=== | ===ASCII=== |
Latest revision as of 18:48, 31 January 2010
A061396
Plain Wiki Table
Large Scale
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.