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

Latest revision as of 18:48, 31 January 2010

A061396

A061396

Plain Wiki Table

Large Scale

\(\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!\)
\(\text{Integer}\!\)	\(\text{Factorization}\!\)	\(\text{Notation}\!\)	\(\text{Riff Digraph}\!\)	\(\text{Rote Graph}\!\)	\(\text{Traversal}\!\)
\(1\!\)	\(1\!\)
\(2\!\)	\(\text{p}_1^1\!\)	\(\text{p}\!\)			\(((~))\)
\(3\!\)	\(\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}\)	\(\text{p}_\text{p}\!\)			\((((~))(~))\)
\(4\!\)	\(\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}\)	\(\text{p}^\text{p}\!\)			\(((((~))))\)
\(5\!\)	\(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \'"`UNIQ-MathJax1-QINU`"' '"`UNIQ-MathJax2-QINU`"' '"`UNIQ-MathJax3-QINU`"' '"`UNIQ-MathJax4-QINU`"' :{\| border="1" cellpadding="20" \| [[Image:Rote 802701 Big.jpg\|330px]] \|} '"`UNIQ-MathJax5-QINU`"' ==='"`UNIQ--h-40--QINU`"'JPEG=== {\| align="center" border="1" cellpadding="6" \| valign="bottom" \| <p>[[Image:Rote 1 Big.jpg\|20px]]</p><br> <p>\(1\!\) \(a(1) ~=~ 0\)	\(\text{p}\!\) \(a(2) ~=~ 1\)	\(\text{p}_\text{p}\!\) \(a(3) ~=~ 2\)	\(\text{p}^\text{p}\!\) \(a(4) ~=~ 2\)	\(\text{p}_{\text{p}_\text{p}}\!\) \(a(5) ~=~ 3\)
\(\text{p} \text{p}_\text{p}\!\) \(a(6) ~=~ 2\)	\(\text{p}_{\text{p}^\text{p}}\!\) \(a(7) ~=~ 3\)	\(\text{p}^{\text{p}_\text{p}}\!\) \(a(8) ~=~ 3\)	\(\text{p}_\text{p}^\text{p}\!\) \(a(9) ~=~ 2\)	\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(10) ~=~ 3\)
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(11) ~=~ 4\)	\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(a(12) ~=~ 2\)	\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(13) ~=~ 3\)	\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(14) ~=~ 3\)	\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(15) ~=~ 3\)
\(\text{p}^{\text{p}^\text{p}}\!\) \(a(16) ~=~ 3\)	\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(17) ~=~ 4\)	\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(18) ~=~ 2\)	\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(19) ~=~ 4\)	\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(20) ~=~ 3\)
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(21) ~=~ 3\)	\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(22) ~=~ 4\)	\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(23) ~=~ 3\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(a(24) ~=~ 3\)	\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(25) ~=~ 3\)
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(26) ~=~ 3\)	\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(27) ~=~ 3\)	\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(28) ~=~ 3\)	\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(29) ~=~ 4\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(30) ~=~ 3\)
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(a(31) ~=~ 5\)	\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(a(32) ~=~ 4\)	\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(33) ~=~ 4\)	\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(34) ~=~ 4\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(35) ~=~ 3\)
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(36) ~=~ 2\)	\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(a(37) ~=~ 3\)	\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(38) ~=~ 4\)	\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(39) ~=~ 3\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(a(40) ~=~ 3\)
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(a(41) ~=~ 4\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(42) ~=~ 3\)	\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(a(43) ~=~ 4\)	\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(44) ~=~ 4\)	\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(45) ~=~ 3\)
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(46) ~=~ 3\)	\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(47) ~=~ 4\)	\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(a(48) ~=~ 3\)	\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(a(49) ~=~ 3\)	\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(50) ~=~ 3\)
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(51) ~=~ 4\)	\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(52) ~=~ 3\)	\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(a(53) ~=~ 4\)	\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(54) ~=~ 3\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(55) ~=~ 4\)
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(56) ~=~ 3\)	\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(57) ~=~ 4\)	\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(58) ~=~ 4\)	\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(a(59) ~=~ 5\)	\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(60) ~=~ 3\)

ASCII

 Comment

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

 Example

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

A111795

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

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.

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

Latest revision as of 18:48, 31 January 2010

Contents

A061396

Plain Wiki Table

Large Scale

ASCII

A111795

JPEG

ASCII

A111800

TeX + JPEG

ASCII

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semantic

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@@ Line 8: / Line 8: @@
 {| 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"
 | <math>\text{Integer}\!</math>
@@ Line 129: / Line 129: @@
 {| 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"
 | <math>\text{Integer}\!</math>
@@ Line 252: / Line 252: @@
 {| 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"
 |
@@ Line 388: / Line 388: @@
 {| 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"
 |
@@ Line 1,327: / Line 1,327: @@
 | [[Image:Riff 49 Big.jpg|65px]]
 | [[Image:Rote 49 Big.jpg|80px]]
-|}
+|-
-|}
+| <math>53\!</math>
+|
-===ASCII===
+<math>\begin{array}{lll}
+\text{p}_{16}^1
-<pre>
+& = & \text{p}_{\text{p}_1^4}^1
- Example
+\\[12pt]
+& = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1
-    * k | natural numbers n such that |riff(n)| = k
+\\[12pt]
-    * 0 | 1;
+& = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1
-    * 1 | 2;
+\end{array}</math>
-    * 2 | 3, 4;
+| <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math>
-    * 3 | 5, 6, 7, 8, 9, 16;
+| [[Image:Riff 53 Big.jpg|90px]]
-    * 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536;
+| [[Image:Rote 53 Big.jpg|90px]]
-    * The natural number values for the riffs with at most 3 pts are as follows (x = root):
+|-
-    * .................o.......o..o.......o
+| <math>64\!</math>
-    * .................|.......^..|.......^
+|
-    * .................v.......|..v.......|
+<math>\begin{array}{lll}
-    * ...........o..o..o....o..o..o..o.o..o
+\text{p}_1^6
-    * ...........|..^..|....|..|..^..|.^..^
+& = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1}
-    * ...........v..|..v....v..v..|..v/...|
+\\[12pt]
-    * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
+& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
-    * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;
+\end{array}</math>
-</pre>
+| <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]]
+|}
+|}
-==A062537==
+===ASCII===
-* [http://oeis.org/wiki/A062537 A062537]
+<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===
@@ Line 1,362: / Line 1,454: @@
 | valign="bottom" |
 <p>&nbsp;</p><br>
-<p>&nbsp;</p><br>
+<p><math>1\!</math></p><br>
 <p><math>a(1) ~=~ 0</math></p>
 | valign="bottom" |
@@ Line 1,641: / Line 1,733: @@
 <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>
@@ Line 1,649: / Line 1,742: @@
 <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>
 |}
@@ Line 2,400: / Line 2,505: @@
 * [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>
-{| align="center" border="1" cellpadding="6"
+: <math>\begin{array}{lllll}
-| valign="bottom" |
-<p>[[Image:Rooted Node Big.jpg|20px]]</p><br>
+& = & 9 \cdot 89189
-<p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
+& = & 2\!:\!2 ~~ 8638\!:\!1
-| valign="bottom" |
+\\
+& = & 2 \cdot 7 \cdot 617
+& = & 1\!:\!1 ~~ 4\!:\!1 ~~ 113\!:\!1
+\\
+&   &
+& = & 30\!:\!1
+\\
+& = & 2 \cdot 3 \cdot 5
+& = & 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1
+\\
+&   &
+& = & 1\!:\!2
+\\
+&   &
+& = & 2\!:\!1
+\\
+&   &
+& = & 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>\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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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" |
 <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 2,742: / Line 2,951: @@
      * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
      * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
      * 7.840858554516122655953405327738 x 10^371.
+ Example
+    * Writing (prime(i))^j as i:j, we have:
+    * 802701 = 2:2 8638:1
+    * 8638 = 1:1 4:1 113:1
+    * 113 = 30:1
+    * 30 = 1:1 2:1 3:1
+    * 4 = 1:2
+    * 3 = 2:1
+    * 2 = 1:1
+    * 1 = { }
+    * So rote(802701) is the graph:
+    *
+    *                           o-o
+    *                           |
+    *                       o-o o-o
+    *                       |   |
+    *               o-o o-o o-o o-o
+    *               |   |   |   |
+    *             o-o   o===o===o-o
+    *             |     |
+    * o-o o-o o-o o-o   o---------o
+    * |   |   |   |     |
+    * o---o   o===o=====o---------o
+    * |       |
+    * O=======O
+    *
+    * Therefore rhig(802701) = 6.
 </pre>