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>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:48px; background:#f0f0ff"
 
|- style="height:48px; background:#f0f0ff"
 
| <math>\text{Integer}\!</math>
 
| <math>\text{Integer}\!</math>
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{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
|+ style="height:24px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:48px; background:#f0f0ff"
 
|- style="height:48px; background:#f0f0ff"
 
| <math>\text{Integer}\!</math>
 
| <math>\text{Integer}\!</math>
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{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:50px; background:#f0f0ff"
 
|- style="height:50px; background:#f0f0ff"
 
|
 
|
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{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:50px; background:#f0f0ff"
 
|- style="height:50px; background:#f0f0ff"
 
|
 
|
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| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
 
| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
 
| [[Image:Riff 64 Big.jpg|65px]]
 
| [[Image:Riff 64 Big.jpg|65px]]
| [[Image:Rote 64 Big.jpg|20px]]
+
| [[Image:Rote 64 Big.jpg|105px]]
 
|-
 
|-
 
| <math>81\!</math>
 
| <math>81\!</math>
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| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
 
| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
 
| [[Image:Riff 81 Big.jpg|65px]]
 
| [[Image:Riff 81 Big.jpg|65px]]
| [[Image:Rote 81 Big.jpg|20px]]
+
| [[Image:Rote 81 Big.jpg|105px]]
 
|-
 
|-
 
| <math>128\!</math>
 
| <math>128\!</math>
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| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
 
| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
 
| [[Image:Riff 128 Big.jpg|90px]]
 
| [[Image:Riff 128 Big.jpg|90px]]
| [[Image:Rote 128 Big.jpg|50px]]
+
| [[Image:Rote 128 Big.jpg|90px]]
 
|-
 
|-
 
| <math>256\!</math>
 
| <math>256\!</math>
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| <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math>
 
| <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math>
 
| [[Image:Riff 256 Big.jpg|90px]]
 
| [[Image:Riff 256 Big.jpg|90px]]
| [[Image:Rote 256 Big.jpg|50px]]
+
| [[Image:Rote 256 Big.jpg|90px]]
 
|-
 
|-
 
| <math>512\!</math>
 
| <math>512\!</math>
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\end{array}</math>
 
\end{array}</math>
 
| <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math>
 
| <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math>
| [[Image:Riff 512 Big.jpg|50px]]
+
| [[Image:Riff 512 Big.jpg|65px]]
| [[Image:Rote 512 Big.jpg|50px]]
+
| [[Image:Rote 512 Big.jpg|105px]]
 
|-
 
|-
 
| <math>65536\!</math>
 
| <math>65536\!</math>
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\end{array}</math>
 
\end{array}</math>
 
| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
 
| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
| [[Image:Riff 65536 Big.jpg|50px]]
+
| [[Image:Riff 65536 Big.jpg|90px]]
| [[Image:Rote 65536 Big.jpg|50px]]
+
| [[Image:Rote 65536 Big.jpg|115px]]
 
|}
 
|}
 
|}
 
|}
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     * ...........v..|..v....v..v..|..v/...|
 
     * ...........v..|..v....v..v..|..v/...|
 
     * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
 
     * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
     * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;  
+
     * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;
 
</pre>
 
</pre>
  
Line 1,454: Line 1,454:
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>&nbsp;</p><br>
 
<p>&nbsp;</p><br>
<p>&nbsp;</p><br>
+
<p><math>1\!</math></p><br>
 
<p><math>a(1) ~=~ 0</math></p>
 
<p><math>a(1) ~=~ 0</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
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<p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
 
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
 
<p><math>a(4) ~=~ 10</math></p>
 
<p><math>a(4) ~=~ 10</math></p>
 +
|-
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Riff 15 Big.jpg|90px]]</p><br>
 
<p>[[Image:Riff 15 Big.jpg|90px]]</p><br>
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<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>a(6) ~=~ 30</math></p>
 
<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>
 
|}
 
|}
  
Line 2,491: Line 2,504:
  
 
* [http://oeis.org/wiki/A109301 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===
 
===JPEG===
Line 2,496: Line 2,553:
 
{| align="center" border="1" cellpadding="6"
 
{| align="center" border="1" cellpadding="6"
 
| valign="bottom" |
 
| valign="bottom" |
<p>[[Image:Rooted Node Big.jpg|20px]]</p><br>
+
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
<p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
+
<p><math>1\!</math></p><br>
 +
<p><math>a(1) ~=~ 0</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 2 Big.jpg|40px]]</p><br>
 
<p>[[Image:Rote 2 Big.jpg|40px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p>
+
<p><math>\text{p}\!</math></p><br>
 +
<p><math>a(2) ~=~ 1</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
 
<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
+
<p><math>\text{p}_\text{p}\!</math></p><br>
 +
<p><math>a(3) ~=~ 2</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
 
<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
+
<p><math>\text{p}^\text{p}\!</math></p><br>
 +
<p><math>a(4) ~=~ 2</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 5 Big.jpg|40px]]</p><br>
 
<p>[[Image:Rote 5 Big.jpg|40px]]</p><br>
<p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(5) ~=~ 3</math></p>
 
|-
 
|-
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_\text{p}\!</math></p><br>
 +
<p><math>a(6) ~=~ 2</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 7 Big.jpg|65px]]</p><br>
 
<p>[[Image:Rote 7 Big.jpg|65px]]</p><br>
<p><math>\begin{array}{l} 4\!:\!1 \\ 7 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br>
 +
<p><math>a(7) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 8 Big.jpg|65px]]</p><br>
 
<p>[[Image:Rote 8 Big.jpg|65px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!3 \\ 8 \end{array}</math></p>
+
<p><math>\text{p}^{\text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(8) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p>
+
<p><math>\text{p}_\text{p}^\text{p}\!</math></p><br>
 +
<p><math>a(9) ~=~ 2</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 10 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 10 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(10) ~=~ 3</math></p>
 
|-
 
|-
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 11 Big.jpg|40px]]</p><br>
 
<p>[[Image:Rote 11 Big.jpg|40px]]</p><br>
<p><math>\begin{array}{l} 5\!:\!1 \\ 11 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
 +
<p><math>a(11) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p>
+
<p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br>
 +
<p><math>a(12) ~=~ 2</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 13 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 13 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 6\!:\!1 \\ 13 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(13) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 14 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 14 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
 +
<p><math>a(14) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 15 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 15 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}</math></p>
+
<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(15) ~=~ 3</math></p>
 
|-
 
|-
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 16 Big.jpg|90px]]</p><br>
 
<p>[[Image:Rote 16 Big.jpg|90px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!4 \\ 16 \end{array}</math></p>
+
<p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br>
 +
<p><math>a(16) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 17 Big.jpg|65px]]</p><br>
 
<p>[[Image:Rote 17 Big.jpg|65px]]</p><br>
<p><math>\begin{array}{l} 7\!:\!1 \\ 17 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
 +
<p><math>a(17) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
 +
<p><math>a(18) ~=~ 2</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 19 Big.jpg|65px]]</p><br>
 
<p>[[Image:Rote 19 Big.jpg|65px]]</p><br>
<p><math>\begin{array}{l} 8\!:\!1 \\ 19 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
 +
<p><math>a(19) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 20 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 20 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}</math></p>
+
<p><math>\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(20) ~=~ 3</math></p>
 
|-
 
|-
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 21 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 21 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}</math></p>
+
<p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
 +
<p><math>a(21) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 22 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 22 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
 +
<p><math>a(22) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 23 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 23 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 9\!:\!1 \\ 23 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
 +
<p><math>a(23) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 24 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 24 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}</math></p>
+
<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br>
 +
<p><math>a(24) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 25 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 25 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 3\!:\!2 \\ 25 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
 +
<p><math>a(25) ~=~ 3</math></p>
 
|-
 
|-
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 26 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 26 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(26) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 27 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 27 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!3 \\ 27 \end{array}</math></p>
+
<p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(27) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 28 Big.jpg|130px]]</p><br>
 
<p>[[Image:Rote 28 Big.jpg|130px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}</math></p>
+
<p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
 +
<p><math>a(28) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 29 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 29 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 10\!:\!1 \\ 29 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
 +
<p><math>a(29) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 30 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 30 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 31 Big.jpg|40px]]</p><br>
 
<p>[[Image:Rote 31 Big.jpg|40px]]</p><br>
<p><math>\begin{array}{l} 11\!:\!1 \\ 31 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br>
 +
<p><math>a(31) ~=~ 5</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 32 Big.jpg|65px]]</p><br>
 
<p>[[Image:Rote 32 Big.jpg|65px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!5 \\ 32 \end{array}</math></p>
+
<p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
 +
<p><math>a(32) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 33 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 33 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 34 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 34 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
 +
<p><math>a(34) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 35 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 35 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
 
<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p>
+
<p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
 +
<p><math>a(36) ~=~ 2</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 37 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 37 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 12\!:\!1 \\ 37 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(37) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 38 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 38 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
 +
<p><math>a(38) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 39 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 39 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 40 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 40 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 41 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 41 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 13\!:\!1 \\ 41 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br>
 +
<p><math>a(41) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 42 Big.jpg|145px]]</p><br>
 
<p>[[Image:Rote 42 Big.jpg|145px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 43 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 43 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 14\!:\!1 \\ 43 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br>
 +
<p><math>a(43) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 44 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 44 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 45 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 45 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 46 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 46 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
 +
<p><math>a(46) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 47 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 47 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 15\!:\!1 \\ 47 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 48 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 48 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}</math></p>
+
<p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br>
 +
<p><math>a(48) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 49 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 49 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 4\!:\!2 \\ 49 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br>
 +
<p><math>a(49) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 50 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 50 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
 +
<p><math>a(50) ~=~ 3</math></p>
 
|-
 
|-
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 51 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 51 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 52 Big.jpg|145px]]</p><br>
 
<p>[[Image:Rote 52 Big.jpg|145px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 53 Big.jpg|90px]]</p><br>
 
<p>[[Image:Rote 53 Big.jpg|90px]]</p><br>
<p><math>\begin{array}{l} 16\!:\!1 \\ 53 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br>
 +
<p><math>a(53) ~=~ 4</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 54 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 54 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}</math></p>
+
<p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
 +
<p><math>a(54) ~=~ 3</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 55 Big.jpg|80px]]</p><br>
 
<p>[[Image:Rote 55 Big.jpg|80px]]</p><br>
<p><math>\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 56 Big.jpg|130px]]</p><br>
 
<p>[[Image:Rote 56 Big.jpg|130px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 57 Big.jpg|105px]]</p><br>
 
<p>[[Image:Rote 57 Big.jpg|105px]]</p><br>
<p><math>\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 58 Big.jpg|120px]]</p><br>
 
<p>[[Image:Rote 58 Big.jpg|120px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}</math></p>
+
<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" |
 
| valign="bottom" |
 
<p>[[Image:Rote 59 Big.jpg|65px]]</p><br>
 
<p>[[Image:Rote 59 Big.jpg|65px]]</p><br>
<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p>
+
<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br>
 +
<p><math>a(59) ~=~ 5</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 60 Big.jpg|155px]]</p><br>
 
<p>[[Image:Rote 60 Big.jpg|155px]]</p><br>
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}</math></p>
+
<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>
 
|}
 
|}
  
Line 2,834: Line 2,951:
 
     * 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>
  

Latest revision as of 18:48, 31 January 2010

A061396

Plain Wiki Table

Large Scale

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

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

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

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

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


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

Rote 2 Big.jpg


\(\text{p}\!\)


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

Rote 3 Big.jpg


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


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

Rote 4 Big.jpg


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


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

Rote 5 Big.jpg


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


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

Rote 6 Big.jpg


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


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

Rote 7 Big.jpg


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


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

Rote 8 Big.jpg


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


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

Rote 9 Big.jpg


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


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

Rote 10 Big.jpg


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


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

Rote 11 Big.jpg


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


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

Rote 12 Big.jpg


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


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

Rote 13 Big.jpg


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


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

Rote 14 Big.jpg


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


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

Rote 15 Big.jpg


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


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

Rote 16 Big.jpg


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


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

Rote 17 Big.jpg


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


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

Rote 18 Big.jpg


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


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

Rote 19 Big.jpg


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


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

Rote 20 Big.jpg


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


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

Rote 21 Big.jpg


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


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

Rote 22 Big.jpg


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


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

Rote 23 Big.jpg


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


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

Rote 24 Big.jpg


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


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

Rote 25 Big.jpg


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


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

Rote 26 Big.jpg


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


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

Rote 27 Big.jpg


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


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

Rote 28 Big.jpg


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


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

Rote 29 Big.jpg


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


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

Rote 30 Big.jpg


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


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

Rote 31 Big.jpg


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


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

Rote 32 Big.jpg


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


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

Rote 33 Big.jpg


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


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

Rote 34 Big.jpg


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


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

Rote 35 Big.jpg


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


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

Rote 36 Big.jpg


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


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

Rote 37 Big.jpg


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


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

Rote 38 Big.jpg


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


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

Rote 39 Big.jpg


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


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

Rote 40 Big.jpg


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


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

Rote 41 Big.jpg


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


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

Rote 42 Big.jpg


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


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

Rote 43 Big.jpg


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


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

Rote 44 Big.jpg


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


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

Rote 45 Big.jpg


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


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

Rote 46 Big.jpg


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


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

Rote 47 Big.jpg


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


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

Rote 48 Big.jpg


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


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

Rote 49 Big.jpg


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


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

Rote 50 Big.jpg


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


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

Rote 51 Big.jpg


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


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

Rote 52 Big.jpg


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


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

Rote 53 Big.jpg


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


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

Rote 54 Big.jpg


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


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

Rote 55 Big.jpg


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


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

Rote 56 Big.jpg


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


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

Rote 57 Big.jpg


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


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

Rote 58 Big.jpg


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


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

Rote 59 Big.jpg


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


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

Rote 60 Big.jpg


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


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

ASCII

 Comment

    * Table of Rotes and Primal Functions for Positive Integers from 1 to 40
    *                                                                        
    *                                                         o-o            
    *                                                         |              
    *                             o-o             o-o         o-o            
    *                             |               |           |              
    *               o-o           o-o           o-o           o-o            
    *               |             |             |             |              
    * O             O             O             O             O              
    *                                                                        
    * { }           1:1           2:1           1:2           3:1            
    *                                                                        
    * 1             2             3             4             5              
    *                                                                        
    *                                                                        
    *                 o-o           o-o                           o-o        
    *                 |             |                             |          
    *     o-o       o-o             o-o         o-o o-o           o-o        
    *     |         |               |           |   |             |          
    * o-o o-o       o-o           o-o           o---o         o-o o-o        
    * |   |         |             |             |             |   |          
    * O===O         O             O             O             O===O          
    *                                                                        
    * 1:1 2:1       4:1           1:3           2:2           1:1 3:1        
    *                                                                        
    * 6             7             8             9             10             
    *                                                                        
    *                                                                        
    * o-o                                                                    
    * |                                                                      
    * o-o                             o-o             o-o         o-o        
    * |                               |               |           |          
    * o-o             o-o o-o     o-o o-o           o-o       o-o o-o        
    * |               |   |       |   |             |         |   |          
    * o-o           o-o   o-o     o===o-o       o-o o-o       o-o o-o        
    * |             |     |       |             |   |         |   |          
    * O             O=====O       O             O===O         O===O          
    *                                                                        
    * 5:1           1:2 2:1       6:1           1:1 4:1       2:1 3:1        
    *                                                                        
    * 11            12            13            14            15             
    *                                                                        
    *                                                                        
    *                 o-o                         o-o                        
    *                 |                           |                          
    *     o-o       o-o                           o-o               o-o      
    *     |         |                             |                 |        
    *   o-o         o-o               o-o o-o   o-o             o-o o-o      
    *   |           |                 |   |     |               |   |        
    * o-o           o-o           o-o o---o     o-o           o-o   o-o      
    * |             |             |   |         |             |     |        
    * O             O             O===O         O             O=====O        
    *                                                                        
    * 1:4           7:1           1:1 2:2       8:1           1:2 3:1        
    *                                                                        
    * 16            17            18            19            20             
    *                                                                        
    *                                                                        
    *                   o-o                                                  
    *                   |                                                    
    *       o-o         o-o       o-o o-o         o-o         o-o            
    *       |           |         |   |           |           |              
    * o-o o-o           o-o       o---o           o-o o-o     o-o o-o        
    * |   |             |         |               |   |       |   |          
    * o-o o-o       o-o o-o       o-o           o-o   o-o     o---o          
    * |   |         |   |         |             |     |       |              
    * O===O         O===O         O             O=====O       O              
    *                                                                        
    * 2:1 4:1       1:1 5:1       9:1           1:3 2:1       3:2            
    *                                                                        
    * 21            22            23            24            25             
    *                                                                        
    *                                                                        
    *                                               o-o                      
    *                                               |                        
    *         o-o       o-o               o-o       o-o               o-o    
    *         |         |                 |         |                 |      
    *     o-o o-o   o-o o-o         o-o o-o     o-o o-o           o-o o-o    
    *     |   |     |   |           |   |       |   |             |   |      
    * o-o o===o-o   o---o         o-o   o-o     o===o-o       o-o o-o o-o    
    * |   |         |             |     |       |             |   |   |      
    * O===O         O             O=====O       O             O===O===O      
    *                                                                        
    * 1:1 6:1       2:3           1:2 4:1       10:1          1:1 2:1 3:1    
    *                                                                        
    * 26            27            28            29            30             
    *                                                                        
    *                                                                        
    * o-o                                                                    
    * |                                                                      
    * o-o             o-o             o-o             o-o                    
    * |               |               |               |                      
    * o-o             o-o             o-o           o-o       o-o   o-o      
    * |               |               |             |         |     |        
    * o-o             o-o         o-o o-o           o-o       o-o o-o        
    * |               |           |   |             |         |   |          
    * o-o           o-o           o-o o-o       o-o o-o       o-o o-o        
    * |             |             |   |         |   |         |   |          
    * O             O             O===O         O===O         O===O          
    *                                                                        
    * 11:1          1:5           2:1 5:1       1:1 7:1       3:1 4:1        
    *                                                                        
    * 31            32            33            34            35             
    *                                                                        
    *                                                                        
    *                                   o-o                                  
    *                                   |                                    
    *                 o-o o-o           o-o             o-o     o-o o-o      
    *                 |   |             |               |       |   |        
    *   o-o o-o o-o o-o   o-o         o-o       o-o o-o o-o     o-o o-o      
    *   |   |   |   |     |           |         |   |   |       |   |        
    * o-o   o---o   o=====o-o     o-o o-o       o-o o===o-o   o-o   o-o      
    * |     |       |             |   |         |   |         |     |        
    * O=====O       O             O===O         O===O         O=====O        
    *                                                                        
    * 1:2 2:2       12:1          1:1 8:1       2:1 6:1       1:3 3:1        
    *                                                                        
    * 36            37            38            39            40             
    *                                                                        
    * In these Figures, "extended lines of identity" like o===o
    * indicate identified nodes and capital O is the root node.
    * The rote height in gammas is found by finding the number
    * of graphs of the following shape between the root and one
    * of the highest nodes of the tree:
    * o--o
    * |
    * o
    * A sequence like this, that can be regarded as a nonnegative integer
    * measure on positive integers, may have as many as 3 other sequences
    * associated with it. Given that the fiber of a function f at n is all
    * the domain elements that map to n, we always have the fiber minimum
    * or minimum inverse function and may also have the fiber cardinality
    * and the fiber maximum or maximum inverse function. For A109301, the
    * minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the
    * first positive integer whose rote height is n, the fiber cardinality
    * is A109300, giving the number of positive integers of rote height n,
    * while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving
    * the last positive integer whose rote height is n, has the following
    * initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36,
    * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
    * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
    * 7.840858554516122655953405327738 x 10^371.

 Example

    * Writing (prime(i))^j as i:j, we have:
    * 802701 = 2:2 8638:1
    * 8638 = 1:1 4:1 113:1
    * 113 = 30:1
    * 30 = 1:1 2:1 3:1
    * 4 = 1:2
    * 3 = 2:1
    * 2 = 1:1
    * 1 = { }
    * So rote(802701) is the graph:
    *                              
    *                           o-o
    *                           |  
    *                       o-o o-o
    *                       |   |  
    *               o-o o-o o-o o-o
    *               |   |   |   |  
    *             o-o   o===o===o-o
    *             |     |          
    * o-o o-o o-o o-o   o---------o
    * |   |   |   |     |          
    * o---o   o===o=====o---------o
    * |       |                    
    * O=======O                    
    *                              
    * Therefore rhig(802701) = 6.

A111795

JPEG

Rooted Node Big.jpg


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

Rote 2 Big.jpg


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

Rote 3 Big.jpg


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

Rote 4 Big.jpg


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

Rote 5 Big.jpg


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

Rote 7 Big.jpg


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

Rote 8 Big.jpg


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

Rote 11 Big.jpg


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

Rote 16 Big.jpg


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

Rote 17 Big.jpg


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

Rote 19 Big.jpg


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

Rote 31 Big.jpg


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

Rote 32 Big.jpg


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

Rote 53 Big.jpg


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

Rote 59 Big.jpg


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

ASCII

 Example

    * Tables of Rotes and Primal Codes for a(1) to a(9)
    *                                                              
    *                                                 o-o          
    *                                                 |            
    *                           o-o     o-o     o-o   o-o       o-o
    *                           |       |       |     |         |  
    *             o-o     o-o   o-o   o-o       o-o   o-o     o-o  
    *             |       |     |     |         |     |       |    
    *       o-o   o-o   o-o     o-o   o-o     o-o     o-o   o-o    
    *       |     |     |       |     |       |       |     |      
    * O     O     O     O       O     O       O       O     O      
    *                                                              
    * { }   1:1   2:1   1:2     3:1   4:1     1:3     5:1   1:4    
    *                                                              
    * 1     2     3     4       5     7       8       11    16     
    *                                                              

A111800

TeX + JPEG

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

Rote 2500 Big.jpg

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

ASCII

 Example

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