Changes

MyWikiBiz, Author Your Legacy — Friday October 31, 2025
Jump to navigationJump to search

User:Jon Awbrey/SEQUENCES (view source)

Revision as of 18:48, 31 January 2010

8,745 bytes added ,  18:48, 31 January 2010
Undo revision 107256 by Jon Awbrey (Talk)
Line 8: Line 8:     
{| 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>
Line 129: Line 129:     
{| 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>
Line 252: Line 252:     
{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
−
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:50px; background:#f0f0ff"
 
|- style="height:50px; background:#f0f0ff"
 
|
 
|
Line 388: Line 388:     
{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
−
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:50px; background:#f0f0ff"
 
|- style="height:50px; background:#f0f0ff"
 
|
 
|
Line 1,030: Line 1,030:  
& |P_k|
 
& |P_k|
 
\\[10pt]
 
\\[10pt]
−
0 & \{ 1 \} & 1 \\
+
0 & \{ 1 \} & 1
−
1 & \{ 2 \} & 1 \\
+
\\
−
2 & \{ 3, 4 \} & 2 \\
+
1 & \{ 2 \} & 1
−
3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 \\
+
\\
 +
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
 
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>
 
\end{array}</math>
Line 1,098: Line 1,102:  
\text{p}_3^1
 
\text{p}_3^1
 
& = & \text{p}_{\text{p}_2^1}^1
 
& = & \text{p}_{\text{p}_2^1}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
 
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
 
\end{array}</math>
 
\end{array}</math>
Line 1,120: Line 1,124:  
\text{p}_4^1
 
\text{p}_4^1
 
& = & \text{p}_{\text{p}_1^2}^1
 
& = & \text{p}_{\text{p}_1^2}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
 
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
 
\end{array}</math>
 
\end{array}</math>
Line 1,132: Line 1,136:  
\text{p}_1^3
 
\text{p}_1^3
 
& = & \text{p}_1^{\text{p}_2^1}
 
& = & \text{p}_1^{\text{p}_2^1}
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
 
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
 
\end{array}</math>
 
\end{array}</math>
Line 1,154: Line 1,158:  
\text{p}_1^4
 
\text{p}_1^4
 
& = & \text{p}_1^{\text{p}_1^2}
 
& = & \text{p}_1^{\text{p}_1^2}
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
 
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
 
\end{array}</math>
 
\end{array}</math>
Line 1,169: Line 1,173:  
\text{p}_1^1 \text{p}_3^1
 
\text{p}_1^1 \text{p}_3^1
 
& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1
 
& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
 
& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
 
\end{array}</math>
 
\end{array}</math>
Line 1,181: Line 1,185:  
\text{p}_5^1
 
\text{p}_5^1
 
& = & \text{p}_{\text{p}_3^1}^1
 
& = & \text{p}_{\text{p}_3^1}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1
 
& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1
 
& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1
 
\end{array}</math>
 
\end{array}</math>
Line 1,205: Line 1,209:  
\text{p}_6^1
 
\text{p}_6^1
 
& = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1
 
& = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1
 
& = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1
 
\end{array}</math>
 
\end{array}</math>
Line 1,217: Line 1,221:  
\text{p}_1^1 \text{p}_4^1
 
\text{p}_1^1 \text{p}_4^1
 
& = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1
 
& = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
 
& = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
 
\end{array}</math>
 
\end{array}</math>
Line 1,229: Line 1,233:  
\text{p}_7^1
 
\text{p}_7^1
 
& = & \text{p}_{\text{p}_4^1}^1
 
& = & \text{p}_{\text{p}_4^1}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1
 
& = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1
−
\\[10pt]
+
\\[12pt]
 
& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
 
& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
 
\end{array}</math>
 
\end{array}</math>
Line 1,237: Line 1,241:  
| [[Image:Riff 17 Big.jpg|90px]]
 
| [[Image:Riff 17 Big.jpg|90px]]
 
| [[Image:Rote 17 Big.jpg|65px]]
 
| [[Image:Rote 17 Big.jpg|65px]]
−
|}
+
|-
−
|}
+
| <math>18\!</math>
−
 
+
|
−
===ASCII===
+
<math>\begin{array}{lll}
−
 
+
\text{p}_1^1 \text{p}_2^2
−
<pre>
+
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
−
Example
+
\end{array}</math>
−
 
+
| <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math>
−
    * k | natural numbers n such that |riff(n)| = k
+
| [[Image:Riff 18 Big.jpg|65px]]
−
    * 0 | 1;
+
| [[Image:Rote 18 Big.jpg|120px]]
−
    * 1 | 2;
+
|-
−
    * 2 | 3, 4;
+
| <math>19\!</math>
−
    * 3 | 5, 6, 7, 8, 9, 16;
+
|
−
    * 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536;
+
<math>\begin{array}{lll}
−
    * The natural number values for the riffs with at most 3 pts are as follows (x = root):
+
\text{p}_8^1
−
    * .................o.......o..o.......o
+
& = & \text{p}_{\text{p}_1^3}^1
−
    * .................|.......^..|.......^
+
\\[12pt]
−
    * .................v.......|..v.......|
+
& = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1
−
    * ...........o..o..o....o..o..o..o.o..o
+
\\[12pt]
−
    * ...........|..^..|....|..|..^..|.^..^
+
& = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1
−
    * ...........v..|..v....v..v..|..v/...|
+
\end{array}</math>
−
    * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
+
| <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math>
−
    * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;
+
| [[Image:Riff 19 Big.jpg|90px]]
−
</pre>
+
| [[Image:Rote 19 Big.jpg|65px]]
−
 
+
|-
−
==A062537==
+
| <math>23\!</math>
−
 
+
|
−
* [http://oeis.org/wiki/A062537 A062537]
+
<math>\begin{array}{lll}
 +
\text{p}_9^1
 +
& = & \text{p}_{\text{p}_2^2}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math>
 +
| [[Image:Riff 23 Big.jpg|65px]]
 +
| [[Image:Rote 23 Big.jpg|80px]]
 +
|-
 +
| <math>25\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_3^2
 +
& = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math>
 +
| [[Image:Riff 25 Big.jpg|65px]]
 +
| [[Image:Rote 25 Big.jpg|80px]]
 +
|-
 +
| <math>27\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_2^3
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 27 Big.jpg|65px]]
 +
| [[Image:Rote 27 Big.jpg|80px]]
 +
|-
 +
| <math>32\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^5
 +
& = & \text{p}_1^{\text{p}_3^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math>
 +
| [[Image:Riff 32 Big.jpg|90px]]
 +
| [[Image:Rote 32 Big.jpg|65px]]
 +
|-
 +
| <math>49\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_4^2
 +
& = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math>
 +
| [[Image:Riff 49 Big.jpg|65px]]
 +
| [[Image:Rote 49 Big.jpg|80px]]
 +
|-
 +
| <math>53\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_{16}^1
 +
& = & \text{p}_{\text{p}_1^4}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 53 Big.jpg|90px]]
 +
| [[Image:Rote 53 Big.jpg|90px]]
 +
|-
 +
| <math>64\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^6
 +
& = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 64 Big.jpg|65px]]
 +
| [[Image:Rote 64 Big.jpg|105px]]
 +
|-
 +
| <math>81\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_2^4
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
 +
| [[Image:Riff 81 Big.jpg|65px]]
 +
| [[Image:Rote 81 Big.jpg|105px]]
 +
|-
 +
| <math>128\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^7
 +
& = & \text{p}_1^{\text{p}_4^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 128 Big.jpg|90px]]
 +
| [[Image:Rote 128 Big.jpg|90px]]
 +
|-
 +
| <math>256\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^8
 +
& = & \text{p}_1^{\text{p}_1^3}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math>
 +
| [[Image:Riff 256 Big.jpg|90px]]
 +
| [[Image:Rote 256 Big.jpg|90px]]
 +
|-
 +
| <math>512\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^9
 +
& = & \text{p}_1^{\text{p}_2^2}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math>
 +
| [[Image:Riff 512 Big.jpg|65px]]
 +
| [[Image:Rote 512 Big.jpg|105px]]
 +
|-
 +
| <math>65536\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^{16}
 +
& = & \text{p}_1^{\text{p}_1^4}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 65536 Big.jpg|90px]]
 +
| [[Image:Rote 65536 Big.jpg|115px]]
 +
|}
 +
|}
   −
===Wiki + TeX + JPEG===
+
===ASCII===
 +
 
 +
<pre>
 +
Example
 +
 
 +
    * k | natural numbers n such that |riff(n)| = k
 +
    * 0 | 1;
 +
    * 1 | 2;
 +
    * 2 | 3, 4;
 +
    * 3 | 5, 6, 7, 8, 9, 16;
 +
    * 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536;
 +
    * The natural number values for the riffs with at most 3 pts are as follows (x = root):
 +
    * .................o.......o..o.......o
 +
    * .................|.......^..|.......^
 +
    * .................v.......|..v.......|
 +
    * ...........o..o..o....o..o..o..o.o..o
 +
    * ...........|..^..|....|..|..^..|.^..^
 +
    * ...........v..|..v....v..v..|..v/...|
 +
    * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
 +
    * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;
 +
</pre>
 +
 
 +
==A062537==
 +
 
 +
* [http://oeis.org/wiki/A062537 A062537]
 +
 
 +
===Wiki + TeX + JPEG===
    
{| align="center" border="1" cellpadding="10"
 
{| align="center" border="1" cellpadding="10"
Line 1,272: 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" |
Line 1,551: Line 1,733:  
<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>
Line 1,559: Line 1,742:  
<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 2,310: Line 2,505:  
* [http://oeis.org/wiki/A109301 A109301]
 
* [http://oeis.org/wiki/A109301 A109301]
   −
===JPEG===
+
===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"
 
{| 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 2,652: 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>
  
Jon Awbrey
12,136

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/105771...107280"

Navigation menu