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

Navigation menu

Page actions

Page actions

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Navigation

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 205: / Line 205: @@
 \end{array}</math>
 | <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
-| <math>\cdots</math>
+| [[Image:Riff 7 Big.jpg|38px]]
 | [[Image:Rote 7 Big.jpg|38px]]
 | <math>(((((~))))(~))</math>
@@ Line 218: / Line 218: @@
 \end{array}</math>
 | <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
-| <math>\cdots</math>
+| [[Image:Riff 8 Big.jpg|38px]]
 | [[Image:Rote 8 Big.jpg|38px]]
 | <math>(((((~))(~))))</math>
@@ Line 229: / Line 229: @@
 \end{array}</math>
 | <math>\text{p}_\text{p}^\text{p}\!</math>
-| <math>\cdots</math>
+| [[Image:Riff 9 Big.jpg|24px]]
 | [[Image:Rote 9 Big.jpg|48px]]
 | <math>(((~))(((~))))</math>
@@ Line 242: / Line 242: @@
 \end{array}</math>
 | <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
-| <math>\cdots</math>
+| [[Image:Riff 16 Big.jpg|38px]]
 | [[Image:Rote 16 Big.jpg|52px]]
 | <math>((((((~))))))</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 478: / Line 478: @@
 \end{array}</math>
 | <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
-| <math>\cdots</math>
+| [[Image:Riff 7 Big.jpg|38px]]
 | [[Image:Rote 7 Big.jpg|38px]]
 | <math>(((((~))))(~))</math>
@@ Line 491: / Line 491: @@
 \end{array}</math>
 | <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
-| <math>\cdots</math>
+| [[Image:Riff 8 Big.jpg|38px]]
 | [[Image:Rote 8 Big.jpg|38px]]
 | <math>(((((~))(~))))</math>
@@ Line 502: / Line 502: @@
 \end{array}</math>
 | <math>\text{p}_\text{p}^\text{p}\!</math>
-| <math>\cdots</math>
+| [[Image:Riff 9 Big.jpg|24px]]
 | [[Image:Rote 9 Big.jpg|48px]]
 | <math>(((~))(((~))))</math>
@@ Line 515: / Line 515: @@
 \end{array}</math>
 | <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
-| <math>\cdots</math>
+| [[Image:Riff 16 Big.jpg|38px]]
 | [[Image:Rote 16 Big.jpg|52px]]
 | <math>((((((~))))))</math>
@@ Line 1,015: / Line 1,015: @@
 </pre>
-==A109300==
+==A062504==
+* [http://oeis.org/wiki/A062504 A062504]
+===TeX Array===
-* [http://oeis.org/wiki/A109300 A109300]
+{| align="center"
+|
+<math>\begin{array}{l|l|r}
+k
+& P_k
+= \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \}
+= \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \}
+& |P_k|
+\\[10pt]
+& \{ 1 \} & 1
+\\
+& \{ 2 \} & 1
+\\
+& \{ 3, 4 \} & 2
+\\
+& \{ 5, 6, 7, 8, 9, 16 \} & 6
+\\
+& \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20
+\end{array}</math>
+|}
 ===JPEG===
-<br>
+{| align="center" border="1" width="90%"
+|+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math>
-{| align="center" border="1" cellpadding="10"
+|- style="height:50px; background:#f0f0ff"
 |
-<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
+{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
-<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
+| width="10%" | <math>\text{Integer}\!</math>
+| width="25%" | <math>\text{Factorization}\!</math>
+| width="15%" | <math>\text{Notation}\!</math>
+| width="25%" | <math>\text{Riff Digraph}\!</math>
+| width="25%" | <math>\text{Rote Graph}\!</math>
+|}
+|-
 |
-<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
+{| cellpadding="12" style="text-align:center; width:100%"
-<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
+| width="10%" | <math>1\!</math>
+| width="25%" | <math>1\!</math>
+| width="15%" | &nbsp;
+| width="25%" | &nbsp;
+| width="25%" | [[Image:Rote 1 Big.jpg|20px]]
+|}
+|-
 |
-<p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
+{| cellpadding="12" style="text-align:center; width:100%"
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
+| width="10%" | <math>2\!</math>
+| width="25%" | <math>\text{p}_1^1\!</math>
+| width="15%" | <math>\text{p}\!</math>
+| width="25%" | [[Image:Riff 2 Big.jpg|20px]]
+| width="25%" | [[Image:Rote 2 Big.jpg|40px]]
+|}
+|-
 |
-<p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
+{| cellpadding="12" style="text-align:center; width:100%"
-<p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p>
+| width="10%" | <math>3\!</math>
+| width="25%" |
+<math>\begin{array}{lll}
+\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
+\end{array}</math>
+| width="15%" | <math>\text{p}_\text{p}\!</math>
+| width="25%" | [[Image:Riff 3 Big.jpg|40px]]
+| width="25%" | [[Image:Rote 3 Big.jpg|40px]]
+|-
+| <math>4\!</math>
 |
-<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
+<math>\begin{array}{lll}
-<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p>
+\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
+\end{array}</math>
+| <math>\text{p}^\text{p}\!</math>
+| [[Image:Riff 4 Big.jpg|40px]]
+| [[Image:Rote 4 Big.jpg|65px]]
+|}
+|-
 |
-<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
+{| cellpadding="12" style="text-align:center; width:100%"
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p>
+| width="10%" | <math>5\!</math>
-|
+| width="25%" |
-<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
+<math>\begin{array}{lll}
-<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p>
+\text{p}_3^1
-|}
+& = & \text{p}_{\text{p}_2^1}^1
+\\[12pt]
-<br>
+& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
+\end{array}</math>
-===ASCII===
+| width="15%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
+| width="25%" | [[Image:Riff 5 Big.jpg|65px]]
-<pre>
+| width="25%" | [[Image:Rote 5 Big.jpg|40px]]
- Example
+|-
+| <math>6\!</math>
-    * Table of Rotes and Primal Functions for Positive Integers of Rote Height 2
+|
-    *
+<math>\begin{array}{lll}
-    * o-o     o-o       o-o   o-o o-o     o-o o-o       o-o o-o     o-o o-o o-o
+\text{p}_1^1 \text{p}_2^1
-    * |       |         |     |   |       |   |         |   |       |   |   |
+& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
-    * o-o   o-o     o-o o-o   o---o     o-o   o-o   o-o o---o     o-o   o---o
+\end{array}</math>
-    * |     |       |   |     |         |     |     |   |         |     |
+| <math>\text{p} \text{p}_{\text{p}}\!</math>
-    * O     O       O===O     O         O=====O     O===O         O=====O
+| [[Image:Riff 6 Big.jpg|65px]]
-    *
+| [[Image:Rote 6 Big.jpg|80px]]
-    * 2:1   1:2     1:1 2:1   2:2       1:2 2:1     1:1 2:2       1:2 2:2
+|-
-    *
+| <math>7\!</math>
-    * 3     4       6         9         12          18            36
+|
-    *
+<math>\begin{array}{lll}
-</pre>
+\text{p}_4^1
+& = & \text{p}_{\text{p}_1^2}^1
-==A109301==
+\\[12pt]
+& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
-* [http://oeis.org/wiki/A109301 A109301]
+\end{array}</math>
+| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
-===JPEG===
+| [[Image:Riff 7 Big.jpg|65px]]
+| [[Image:Rote 7 Big.jpg|65px]]
-<br>
+|-
+| <math>8\!</math>
-{| align="center" border="1" cellpadding="6"
+|
-| valign="bottom" |
+<math>\begin{array}{lll}
-<p>[[Image:Rooted Node Big.jpg|20px]]</p><br>
+\text{p}_1^3
-<p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
+& = & \text{p}_1^{\text{p}_2^1}
-| valign="bottom" |
+\\[12pt]
-<p>[[Image:Rote 2 Big.jpg|40px]]</p><br>
+& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
-<p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p>
+\end{array}</math>
-| valign="bottom" |
+| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
-<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
+| [[Image:Riff 8 Big.jpg|65px]]
-<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
+| [[Image:Rote 8 Big.jpg|65px]]
-| valign="bottom" |
+|-
-<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
+| <math>9\!</math>
-<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
+|
-| valign="bottom" |
+<math>\begin{array}{lll}
-<p>[[Image:Rote 5 Big.jpg|40px]]</p><br>
+\text{p}_2^2
-<p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
+& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
+\end{array}</math>
+| <math>\text{p}_\text{p}^\text{p}\!</math>
+| [[Image:Riff 9 Big.jpg|40px]]
+| [[Image:Rote 9 Big.jpg|80px]]
 |-
-| valign="bottom" |
+| <math>16\!</math>
-<p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
+|
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_1^4
-<p>[[Image:Rote 7 Big.jpg|65px]]</p><br>
+& = & \text{p}_1^{\text{p}_1^2}
-<p><math>\begin{array}{l} 4\!:\!1 \\ 7 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
-<p>[[Image:Rote 8 Big.jpg|65px]]</p><br>
+\end{array}</math>
-<p><math>\begin{array}{l} 1\!:\!3 \\ 8 \end{array}</math></p>
+| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
-| valign="bottom" |
+| [[Image:Riff 16 Big.jpg|65px]]
-<p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
+| [[Image:Rote 16 Big.jpg|90px]]
-<p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</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>
 |-
-| valign="bottom" |
+|
-<p>[[Image:Rote 11 Big.jpg|40px]]</p><br>
+{| cellpadding="12" style="text-align:center; width:100%"
-<p><math>\begin{array}{l} 5\!:\!1 \\ 11 \end{array}</math></p>
+| width="10%" | <math>10\!</math>
-| valign="bottom" |
+| width="25%" |
-<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
+<math>\begin{array}{lll}
-<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p>
+\text{p}_1^1 \text{p}_3^1
-| valign="bottom" |
+& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1
-<p>[[Image:Rote 13 Big.jpg|80px]]</p><br>
+\\[12pt]
-<p><math>\begin{array}{l} 6\!:\!1 \\ 13 \end{array}</math></p>
+& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
-| valign="bottom" |
+\end{array}</math>
-<p>[[Image:Rote 14 Big.jpg|105px]]</p><br>
+| width="15%" | <math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math>
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}</math></p>
+| width="25%" | [[Image:Riff 10 Big.jpg|90px]]
-| valign="bottom" |
+| width="25%" | [[Image:Rote 10 Big.jpg|80px]]
-<p>[[Image:Rote 15 Big.jpg|80px]]</p><br>
-<p><math>\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}</math></p>
 |-
-| valign="bottom" |
+| <math>11\!</math>
-<p>[[Image:Rote 16 Big.jpg|90px]]</p><br>
+|
-<p><math>\begin{array}{l} 1\!:\!4 \\ 16 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_5^1
-<p>[[Image:Rote 17 Big.jpg|65px]]</p><br>
+& = & \text{p}_{\text{p}_3^1}^1
-<p><math>\begin{array}{l} 7\!:\!1 \\ 17 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1
-<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
+\\[12pt]
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p>
+& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1
-| valign="bottom" |
+\end{array}</math>
-<p>[[Image:Rote 19 Big.jpg|65px]]</p><br>
+| <math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math>
-<p><math>\begin{array}{l} 8\!:\!1 \\ 19 \end{array}</math></p>
+| [[Image:Riff 11 Big.jpg|90px]]
-| valign="bottom" |
+| [[Image:Rote 11 Big.jpg|40px]]
-<p>[[Image:Rote 20 Big.jpg|105px]]</p><br>
+|-
-<p><math>\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}</math></p>
+| <math>12\!</math>
+|
+<math>\begin{array}{lll}
+\text{p}_1^2 \text{p}_2^1
+& = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1
+\end{array}</math>
+| <math>\text{p}^{\text{p}} \text{p}_{\text{p}}\!</math>
+| [[Image:Riff 12 Big.jpg|65px]]
+| [[Image:Rote 12 Big.jpg|105px]]
 |-
-| valign="bottom" |
+| <math>13\!</math>
-<p>[[Image:Rote 21 Big.jpg|105px]]</p><br>
+|
-<p><math>\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_6^1
-<p>[[Image:Rote 22 Big.jpg|80px]]</p><br>
+& = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1
-<p>[[Image:Rote 23 Big.jpg|80px]]</p><br>
+\end{array}</math>
-<p><math>\begin{array}{l} 9\!:\!1 \\ 23 \end{array}</math></p>
+| <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math>
-| valign="bottom" |
+| [[Image:Riff 13 Big.jpg|65px]]
-<p>[[Image:Rote 24 Big.jpg|105px]]</p><br>
+| [[Image:Rote 13 Big.jpg|80px]]
-<p><math>\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}</math></p>
+|-
-| valign="bottom" |
+| <math>14\!</math>
-<p>[[Image:Rote 25 Big.jpg|80px]]</p><br>
+|
-<p><math>\begin{array}{l} 3\!:\!2 \\ 25 \end{array}</math></p>
+<math>\begin{array}{lll}
+\text{p}_1^1 \text{p}_4^1
+& = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1
+\\[12pt]
+& = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
+\end{array}</math>
+| <math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math>
+| [[Image:Riff 14 Big.jpg|90px]]
+| [[Image:Rote 14 Big.jpg|105px]]
 |-
-| valign="bottom" |
+| <math>17\!</math>
-<p>[[Image:Rote 26 Big.jpg|120px]]</p><br>
+|
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_7^1
-<p>[[Image:Rote 27 Big.jpg|80px]]</p><br>
+& = & \text{p}_{\text{p}_4^1}^1
-<p><math>\begin{array}{l} 2\!:\!3 \\ 27 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1
-<p>[[Image:Rote 28 Big.jpg|130px]]</p><br>
+\\[12pt]
-<p><math>\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}</math></p>
+& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
-| valign="bottom" |
+\end{array}</math>
-<p>[[Image:Rote 29 Big.jpg|80px]]</p><br>
+| <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math>
-<p><math>\begin{array}{l} 10\!:\!1 \\ 29 \end{array}</math></p>
+| [[Image:Riff 17 Big.jpg|90px]]
-| valign="bottom" |
+| [[Image:Rote 17 Big.jpg|65px]]
-<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>
 |-
-| valign="bottom" |
+| <math>18\!</math>
-<p>[[Image:Rote 31 Big.jpg|40px]]</p><br>
+|
-<p><math>\begin{array}{l} 11\!:\!1 \\ 31 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_1^1 \text{p}_2^2
-<p>[[Image:Rote 32 Big.jpg|65px]]</p><br>
+& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
-<p><math>\begin{array}{l} 1\!:\!5 \\ 32 \end{array}</math></p>
+\end{array}</math>
-| valign="bottom" |
+| <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math>
-<p>[[Image:Rote 33 Big.jpg|80px]]</p><br>
+| [[Image:Riff 18 Big.jpg|65px]]
-<p><math>\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}</math></p>
+| [[Image:Rote 18 Big.jpg|120px]]
-| 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>
-| 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>
 |-
-| valign="bottom" |
+| <math>19\!</math>
-<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
+|
-<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_8^1
-<p>[[Image:Rote 37 Big.jpg|105px]]</p><br>
+& = & \text{p}_{\text{p}_1^3}^1
-<p><math>\begin{array}{l} 12\!:\!1 \\ 37 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1
-<p>[[Image:Rote 38 Big.jpg|105px]]</p><br>
+\\[12pt]
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}</math></p>
+& = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1
-| valign="bottom" |
+\end{array}</math>
-<p>[[Image:Rote 39 Big.jpg|120px]]</p><br>
+| <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math>
-<p><math>\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}</math></p>
+| [[Image:Riff 19 Big.jpg|90px]]
-| valign="bottom" |
+| [[Image:Rote 19 Big.jpg|65px]]
-<p>[[Image:Rote 40 Big.jpg|105px]]</p><br>
-<p><math>\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}</math></p>
 |-
-| valign="bottom" |
+| <math>23\!</math>
-<p>[[Image:Rote 41 Big.jpg|80px]]</p><br>
+|
-<p><math>\begin{array}{l} 13\!:\!1 \\ 41 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_9^1
-<p>[[Image:Rote 42 Big.jpg|145px]]</p><br>
+& = & \text{p}_{\text{p}_2^2}^1
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1
-<p>[[Image:Rote 43 Big.jpg|105px]]</p><br>
+\end{array}</math>
-<p><math>\begin{array}{l} 14\!:\!1 \\ 43 \end{array}</math></p>
+| <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math>
-| valign="bottom" |
+| [[Image:Riff 23 Big.jpg|65px]]
-<p>[[Image:Rote 44 Big.jpg|105px]]</p><br>
+| [[Image:Rote 23 Big.jpg|80px]]
-<p><math>\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}</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>
 |-
-| valign="bottom" |
+| <math>25\!</math>
-<p>[[Image:Rote 46 Big.jpg|120px]]</p><br>
+|
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_3^2
-<p>[[Image:Rote 47 Big.jpg|80px]]</p><br>
+& = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1}
-<p><math>\begin{array}{l} 15\!:\!1 \\ 47 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1}
-<p>[[Image:Rote 48 Big.jpg|105px]]</p><br>
+\end{array}</math>
-<p><math>\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}</math></p>
+| <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math>
-| valign="bottom" |
+| [[Image:Riff 25 Big.jpg|65px]]
-<p>[[Image:Rote 49 Big.jpg|80px]]</p><br>
+| [[Image:Rote 25 Big.jpg|80px]]
-<p><math>\begin{array}{l} 4\!:\!2 \\ 49 \end{array}</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>
 |-
-| valign="bottom" |
+| <math>27\!</math>
-<p>[[Image:Rote 51 Big.jpg|105px]]</p><br>
+|
-<p><math>\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_2^3
-<p>[[Image:Rote 52 Big.jpg|145px]]</p><br>
+& = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1}
-<p><math>\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1}
-<p>[[Image:Rote 53 Big.jpg|90px]]</p><br>
+\end{array}</math>
-<p><math>\begin{array}{l} 16\!:\!1 \\ 53 \end{array}</math></p>
+| <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math>
-| valign="bottom" |
+| [[Image:Riff 27 Big.jpg|65px]]
-<p>[[Image:Rote 54 Big.jpg|120px]]</p><br>
+| [[Image:Rote 27 Big.jpg|80px]]
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}</math></p>
+|-
-| valign="bottom" |
+| <math>32\!</math>
-<p>[[Image:Rote 55 Big.jpg|80px]]</p><br>
+|
-<p><math>\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}</math></p>
+<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]]
 |-
-| valign="bottom" |
+| <math>53\!</math>
-<p>[[Image:Rote 56 Big.jpg|130px]]</p><br>
+|
-<p><math>\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}</math></p>
+<math>\begin{array}{lll}
-| valign="bottom" |
+\text{p}_{16}^1
-<p>[[Image:Rote 57 Big.jpg|105px]]</p><br>
+& = & \text{p}_{\text{p}_1^4}^1
-<p><math>\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}</math></p>
+\\[12pt]
-| valign="bottom" |
+& = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1
-<p>[[Image:Rote 58 Big.jpg|120px]]</p><br>
+\\[12pt]
-<p><math>\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}</math></p>
+& = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1
-| valign="bottom" |
+\end{array}</math>
-<p>[[Image:Rote 59 Big.jpg|65px]]</p><br>
+| <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math>
-<p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p>
+| [[Image:Riff 53 Big.jpg|90px]]
-| valign="bottom" |
+| [[Image:Rote 53 Big.jpg|90px]]
-<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>
+| <math>64\!</math>
-|}
+|
+<math>\begin{array}{lll}
-<br>
+\text{p}_1^6
+& = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1}
-===ASCII===
+\\[12pt]
+& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
-<pre>
+\end{array}</math>
-  Comment
+| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
+| [[Image:Riff 64 Big.jpg|65px]]
-     * Table of Rotes and Primal Functions for Positive Integers from 1 to 40
+| [[Image:Rote 64 Big.jpg|105px]]
-     *
+|-
-     *                                                         o-o
+| <math>81\!</math>
-     *                                                         |
+|
-     *                             o-o             o-o         o-o
+<math>\begin{array}{lll}
-     *                             |               |           |
+\text{p}_2^4
-     *               o-o           o-o           o-o           o-o
+& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2}
-     *               |             |             |             |
+\\[12pt]
-     * O             O             O             O             O
+& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}}
-     *
+\end{array}</math>
-     * { }           1:1           2:1           1:2           3:1
+| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
-     *
+| [[Image:Riff 81 Big.jpg|65px]]
-     * 1             2             3             4             5
+| [[Image:Rote 81 Big.jpg|105px]]
-     *
+|-
-     *
+| <math>128\!</math>
-     *                 o-o           o-o                           o-o
+|
-     *                 |             |                             |
+<math>\begin{array}{lll}
-     *     o-o       o-o             o-o         o-o o-o           o-o
+\text{p}_1^7
-     *     |         |               |           |   |             |
+& = & \text{p}_1^{\text{p}_4^1}
-     * o-o o-o       o-o           o-o           o---o         o-o o-o
+\\[12pt]
-     * |   |         |             |             |             |   |
+& = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1}
-     * O===O         O             O             O             O===O
+\\[12pt]
-     *
+& = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}
-     * 1:1 2:1       4:1           1:3           2:2           1:1 3:1
+\end{array}</math>
-     *
+| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
-     * 6             7             8             9             10
+| [[Image:Riff 128 Big.jpg|90px]]
-     *
+| [[Image:Rote 128 Big.jpg|90px]]
-     *
+|-
-     * o-o
+| <math>256\!</math>
-     * |
+|
-     * o-o                             o-o             o-o         o-o
+<math>\begin{array}{lll}
-     * |                               |               |           |
+\text{p}_1^8
-     * o-o             o-o o-o     o-o o-o           o-o       o-o o-o
+& = & \text{p}_1^{\text{p}_1^3}
-     * |               |   |       |   |             |         |   |
+\\[12pt]
-     * o-o           o-o   o-o     o===o-o       o-o o-o       o-o o-o
+& = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}}
-     * |             |     |       |             |   |         |   |
+\\[12pt]
-     * O             O=====O       O             O===O         O===O
+& = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}
-     *
+\end{array}</math>
-     * 5:1           1:2 2:1       6:1           1:1 4:1       2:1 3:1
+| <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math>
-     *
+| [[Image:Riff 256 Big.jpg|90px]]
-     * 11            12            13            14            15
+| [[Image:Rote 256 Big.jpg|90px]]
-     *
+|-
-     *
+| <math>512\!</math>
-     *                 o-o                         o-o
+|
-     *                 |                           |
+<math>\begin{array}{lll}
-     *     o-o       o-o                           o-o               o-o
+\text{p}_1^9
-     *     |         |                             |                 |
+& = & \text{p}_1^{\text{p}_2^2}
-     *   o-o         o-o               o-o o-o   o-o             o-o o-o
+\\[12pt]
-     *   |           |                 |   |     |               |   |
+& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}
-     * o-o           o-o           o-o o---o     o-o           o-o   o-o
+\end{array}</math>
-     * |             |             |   |         |             |     |
+| <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math>
-     * O             O             O===O         O             O=====O
+| [[Image:Riff 512 Big.jpg|65px]]
-     *
+| [[Image:Rote 512 Big.jpg|105px]]
-     * 1:4           7:1           1:1 2:2       8:1           1:2 3:1
+|-
-     *
+| <math>65536\!</math>
-     * 16            17            18            19            20
+|
-     *
+<math>\begin{array}{lll}
-     *
+\text{p}_1^{16}
-     *                   o-o
+& = & \text{p}_1^{\text{p}_1^4}
-     *                   |
+\\[12pt]
-     *       o-o         o-o       o-o o-o         o-o         o-o
+& = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}}
-     *       |           |         |   |           |           |
+\\[12pt]
-     * o-o o-o           o-o       o---o           o-o o-o     o-o o-o
+& = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}
-     * |   |             |         |               |   |       |   |
+\end{array}</math>
-     * o-o o-o       o-o o-o       o-o           o-o   o-o     o---o
+| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
-     * |   |         |   |         |             |     |       |
+| [[Image:Riff 65536 Big.jpg|90px]]
-     * O===O         O===O         O             O=====O       O
+| [[Image:Rote 65536 Big.jpg|115px]]
-     *
+|}
-     * 2:1 4:1       1:1 5:1       9:1           1:3 2:1       3:2
+|}
-     *
-     * 21            22            23            24            25
+===ASCII===
-     *
-     *
+<pre>
-     *                                               o-o
+ Example
-     *                                               |
-     *         o-o       o-o               o-o       o-o               o-o
+    * k | natural numbers n such that |riff(n)| = k
-     *         |         |                 |         |                 |
+    * 0 | 1;
-     *     o-o o-o   o-o o-o         o-o o-o     o-o o-o           o-o o-o
+    * 1 | 2;
-     *     |   |     |   |           |   |       |   |             |   |
+    * 2 | 3, 4;
-     * o-o o===o-o   o---o         o-o   o-o     o===o-o       o-o o-o o-o
+    * 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;
-     * O===O         O             O=====O       O             O===O===O
+    * The natural number values for the riffs with at most 3 pts are as follows (x = root):
-     *
+    * .................o.......o..o.......o
-     * 1:1 6:1       2:3           1:2 4:1       10:1          1:1 2:1 3:1
+    * .................|.......^..|.......^
-     *
+    * .................v.......|..v.......|
-     * 26            27            28            29            30
+    * ...........o..o..o....o..o..o..o.o..o
-     *
+    * ...........|..^..|....|..|..^..|.^..^
-     *
+    * ...........v..|..v....v..v..|..v/...|
-     * o-o
+    * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
-     * |
+    * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;
-     * o-o             o-o             o-o             o-o
+</pre>
-     * |               |               |               |
-     * o-o             o-o             o-o           o-o       o-o   o-o
+==A062537==
-     * |               |               |             |         |     |
-     * o-o             o-o         o-o o-o           o-o       o-o o-o
+* [http://oeis.org/wiki/A062537 A062537]
-     * |               |           |   |             |         |   |
-     * o-o           o-o           o-o o-o       o-o o-o       o-o o-o
+===Wiki + TeX + JPEG===
-     * |             |             |   |         |   |         |   |
-     * O             O             O===O         O===O         O===O
+{| align="center" border="1" cellpadding="10"
-     *
+|+ style="height:25px" | <math>a(n) = \text{Number of Nodes in the Riff of}~ n</math>
-     * 11:1          1:5           2:1 5:1       1:1 7:1       3:1 4:1
+| valign="bottom" |
-     *
+<p>&nbsp;</p><br>
-     * 31            32            33            34            35
+<p><math>1\!</math></p><br>
-     *
+<p><math>a(1) ~=~ 0</math></p>
-     *
+| valign="bottom" |
-     *                                   o-o
+<p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
-     *                                   |
+<p><math>\text{p}\!</math></p><br>
-     *                 o-o o-o           o-o             o-o     o-o o-o
+<p><math>a(2) ~=~ 1</math></p>
-     *                 |   |             |               |       |   |
+| valign="bottom" |
-     *   o-o o-o o-o o-o   o-o         o-o       o-o o-o o-o     o-o o-o
+<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
-     *   |   |   |   |     |           |         |   |   |       |   |
+<p><math>\text{p}_\text{p}\!</math></p><br>
-     * o-o   o---o   o=====o-o     o-o o-o       o-o o===o-o   o-o   o-o
+<p><math>a(3) ~=~ 2</math></p>
-     * |     |       |             |   |         |   |         |     |
+| valign="bottom" |
-     * O=====O       O             O===O         O===O         O=====O
+<p>[[Image:Riff 4 Big.jpg|40px]]</p><br>
-     *
+<p><math>\text{p}^\text{p}\!</math></p><br>
+<p><math>a(4) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 5 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(5) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 6 Big.jpg|65px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}}\!</math></p><br>
+<p><math>a(6) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 7 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}}}\!</math></p><br>
+<p><math>a(7) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 8 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(8) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 9 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>a(9) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 10 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(10) ~=~ 4</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 11 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!</math></p><br>
+<p><math>a(11) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 12 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br>
+<p><math>a(12) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 13 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(13) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 14 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math></p><br>
+<p><math>a(14) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 15 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(15) ~=~ 5</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 16 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}^{\text{p}}}\!</math></p><br>
+<p><math>a(16) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 17 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math></p><br>
+<p><math>a(17) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 18 Big.jpg|65px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>a(18) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 19 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math></p><br>
+<p><math>a(19) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 20 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(20) ~=~ 5</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 21 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(21) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 22 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(22) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 23 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(23) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 24 Big.jpg|115px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br>
+<p><math>a(24) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 25 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
+<p><math>a(25) ~=~ 4</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 26 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(26) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 27 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(27) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 28 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(28) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 29 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(29) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 30 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(30) ~=~ 6</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 31 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br>
+<p><math>a(31) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 32 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(32) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 33 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(33) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 34 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(34) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 35 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(35) ~=~ 6</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 36 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>a(36) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 37 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(37) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 38 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(38) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 39 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(39) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 40 Big.jpg|135px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(40) ~=~ 6</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 41 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(41) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 42 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(42) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 43 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(43) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 44 Big.jpg|115px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(44) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 45 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(45) ~=~ 6</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 46 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(46) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 47 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(47) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 48 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br>
+<p><math>a(48) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 49 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br>
+<p><math>a(49) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 50 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
+<p><math>a(50) ~=~ 5</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 51 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(51) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 52 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(52) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 53 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(53) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 54 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(54) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 55 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(55) ~=~ 7</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 56 Big.jpg|135px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(56) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 57 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(57) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 58 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(58) ~=~ 6</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 59 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br>
+<p><math>a(59) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 60 Big.jpg|115px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(60) ~=~ 7</math></p>
+|}
+==A062860==
+* [http://oeis.org/wiki/A062860 A062860]
+===Wiki + TeX + JPEG===
+{| align="center" border="1" cellpadding="10"
+|+ style="height:25px" | <math>a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}</math>
+| valign="bottom" |
+<p>&nbsp;</p><br>
+<p><math>1\!</math></p><br>
+<p><math>a(0) ~=~ 1</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
+<p><math>\text{p}\!</math></p><br>
+<p><math>a(1) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_\text{p}\!</math></p><br>
+<p><math>a(2) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 5 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(3) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 10 Big.jpg|90px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(4) ~=~ 10</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Riff 15 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
+<p><math>a(5) ~=~ 15</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 30 Big.jpg|115px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(6) ~=~ 30</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 55 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(7) ~=~ 55</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 105 Big.jpg|115px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(8) ~=~ 105</math></p>
+| valign="bottom" |
+<p>[[Image:Riff 165 Big.jpg|135px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(9) ~=~ 165</math></p>
+|}
+==A106177==
+* [http://oeis.org/wiki/A106177 A106177]
+===Primal Codes of Finite Partial Functions on Positive Integers===
+{| align="center"
+|
+<math>\begin{array}{rcl}
+& = & \varnothing \\
+& = & 1\!:\!1 \\
+& = & 2\!:\!1 \\
+& = & 1\!:\!2 \\
+& = & 3\!:\!1 \\
+& = & 1\!:\!1 ~~ 2\!:\!1 \\
+& = & 4\!:\!1 \\
+& = & 1\!:\!3 \\
+& = & 2\!:\!2 \\
+& = & 1\!:\!1 ~~ 3\!:\!1 \\
+& = & 5\!:\!1 \\
+& = & 1\!:\!2 ~~ 2\!:\!1 \\
+& = & 6\!:\!1 \\
+& = & 1\!:\!1 ~~ 4\!:\!1 \\
+& = & 2\!:\!1 ~~ 3\!:\!1 \\
+& = & 1\!:\!4 \\
+& = & 7\!:\!1 \\
+& = & 1\!:\!1 ~~ 2\!:\!2 \\
+& = & 8\!:\!1 \\
+& = & 1\!:\!2 ~~ 3\!:\!1
+\end{array}</math>
+|}
+===Wiki Table===
+{| align="center" style="font-weight:bold; text-align:center"
+| || || || || || || || ||
+| <font color="red">1</font>
+|
+| <font color="red">1</font>
+|-
+| || || || || || || ||
+| <font color="red">2</font>
+| || 1 ||
+| <font color="red">2</font>
+|-
+| || || || || || ||
+| <font color="red">3</font>
+| || 1 || || 1 ||
+| <font color="red">3</font>
+|-
+| || || || || ||
+| <font color="red">4</font>
+| || 1 || || 2 || || 1 ||
+| <font color="red">4</font>
+|-
+| || || || ||
+| <font color="red">5</font>
+| || 1 || || 3 || || 1 || || 1 ||
+| <font color="red">5</font>
+|-
+| || || ||
+| <font color="red">6</font>
+| || 1 || || 1 || || 1 || || 4 || || 1 ||
+| <font color="red">6</font>
+|-
+| || ||
+| <font color="red">7</font>
+| || 1 || || 5 || || 2 || || 9 || || 1 || || 1 ||
+| <font color="red">7</font>
+|-
+| ||
+| <font color="red">8</font>
+| || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">8</font>
+|-
+|
+| <font color="red">9</font>
+| || 1 || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || 1 ||
+| <font color="red">9</font>
+|-
+| width="12pt" | <font color="red">10</font>
+| width="12pt" |
+| width="12pt" | 1
+| width="12pt" |
+| width="12pt" | 1
+| width="12pt" |
+| width="12pt" | 1
+| width="12pt" |
+| width="12pt" | 36
+| width="12pt" |
+| width="12pt" | 1
+| width="12pt" |
+| width="12pt" | 2
+| width="12pt" |
+| width="12pt" | 1
+| width="12pt" |
+| width="12pt" | 8
+| width="12pt" |
+| width="12pt" | 1
+| width="12pt" |
+| width="12pt" | <font color="red">10</font>
+|}
+===Wiki + TeX===
+====Smallmatrix====
+{| align="center"
+|
+<math>\begin{smallmatrix}
+& & & & & & & & & {\color{red}1} & & {\color{red}1}
+\\
+& & & & & & & & {\color{red}2} & & 1 & & {\color{red}2}
+\\
+& & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3}
+\\
+& & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4}
+\\
+& & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5}
+\\
+& & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6}
+\\
+& & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7}
+\\
+& & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8}
+\\
+& {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9}
+\\
+{\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10}
+\end{smallmatrix}</math>
+|}
+====Array====
+{| align="center"
+|
+<math>\begin{array}{*{21}{c}}
+& & & & & & & & & {\color{red}1} & & {\color{red}1}
+\\
+& & & & & & & & {\color{red}2} & & 1 & & {\color{red}2}
+\\
+& & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3}
+\\
+& & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4}
+\\
+& & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5}
+\\
+& & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6}
+\\
+& & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7}
+\\
+& & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8}
+\\
+& {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9}
+\\
+{\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10}
+\end{array}</math>
+|}
+====Matrix====
+{| align="center"
+|
+<math>\begin{matrix}
+n \circ m
+\\
+~/~\backslash~ 1
+\\
+~/~ 1 ~\backslash~ 2
+\\
+~/~ 1 \cdot 1 ~\backslash~ 3
+\\
+~/~ 1 \cdot 2 \cdot 1 ~\backslash~ 4
+\\
+~/~ 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 5
+\\
+~/~ 1 \cdot 1 \cdot 1 \cdot 4 \cdot 1 ~\backslash~ 6
+\\
+~/~ 1 \cdot 5 \cdot 2 \cdot 9 \cdot 1 \cdot 1 ~\backslash~ 7
+\\
+~/~ 1 \cdot 6 \cdot 1 \cdot 1 \cdot 1 \cdot 2 \cdot 1 ~\backslash~ 8
+\\
+~/~ 1 \cdot 7 \cdot 1 \cdot 25\cdot 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 9
+\\
+~/~ 1 \cdot 1 \cdot 1 \cdot 36\cdot 1 \cdot 2 \cdot 1 \cdot 8 \cdot 1 ~\backslash~ 10
+\end{matrix}</math>
+|}
+===ASCII===
+<pre>
+ Example
+    *                      n o m
+    *                       \ /
+    *                      1 . 1
+    *                     \ / \ /
+    *                    2 . 1 . 2
+    *                   \ / \ / \ /
+    *                  3 . 1 . 1 . 3
+    *                 \ / \ / \ / \ /
+    *                4 . 1 . 2 . 1 . 4
+    *               \ / \ / \ / \ / \ /
+    *              5 . 1 . 3 . 1 . 1 . 5
+    *             \ / \ / \ / \ / \ / \ /
+    *            6 . 1 . 1 . 1 . 4 . 1 . 6
+    *           \ / \ / \ / \ / \ / \ / \ /
+    *          7 . 1 . 5 . 2 . 9 . 1 . 1 . 7
+    *         \ / \ / \ / \ / \ / \ / \ / \ /
+    *        8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8
+    *       \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *      9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9
+    *     \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *   10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10
+    *
+    * Primal codes of finite partial functions on positive integers:
+    * 1 = { }
+    * 2 = 1:1
+    * 3 = 2:1
+    * 4 = 1:2
+    * 5 = 3:1
+    * 6 = 1:1 2:1
+    * 7 = 4:1
+    * 8 = 1:3
+    * 9 = 2:2
+    * 10 = 1:1 3:1
+    * 11 = 5:1
+    * 12 = 1:2 2:1
+    * 13 = 6:1
+    * 14 = 1:1 4:1
+    * 15 = 2:1 3:1
+    * 16 = 1:4
+    * 17 = 7:1
+    * 18 = 1:1 2:2
+    * 19 = 8:1
+    * 20 = 1:2 3:1
+</pre>
+==A106178==
+* [http://oeis.org/wiki/A106178 A106178]
+===Wiki Table===
+{| align="center" style="font-weight:bold; text-align:center; width:90%"
+| || || || || || || || || || || || || || ||
+| <font color="red">1</font>
+|
+| <font color="red">1</font>
+|-
+| || || || || || || || || || || || || ||
+| <font color="red">2</font>
+| || &middot; ||
+| <font color="red">2</font>
+|-
+| || || || || || || || || || || || ||
+| <font color="red">3</font>
+| || &middot; || || &middot; ||
+| <font color="red">3</font>
+|-
+| || || || || || || || || || || ||
+| <font color="red">4</font>
+| || &middot; || || 2 || || &middot; ||
+| <font color="red">4</font>
+|-
+| || || || || || || || || || ||
+| <font color="red">5</font>
+| || &middot; || || 3 || || 1 || || &middot; ||
+| <font color="red">5</font>
+|-
+| || || || || || || || || ||
+| <font color="red">6</font>
+| || &middot; || || 1 || || 1 || || 4 || || &middot; ||
+| <font color="red">6</font>
+|-
+| || || || || || || || ||
+| <font color="red">7</font>
+| || &middot; || || 5 || || 2 || || 9 || || 1 || || &middot; ||
+| <font color="red">7</font>
+|-
+| || || || || || || ||
+| <font color="red">8</font>
+| || &middot; || || 6 || || 1 || || 1 || || 1 || || 2 || || &middot; ||
+| <font color="red">8</font>
+|-
+| || || || || || ||
+| <font color="red">9</font>
+| || &middot; || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || &middot; ||
+| <font color="red">9</font>
+|-
+| || || || || ||
+| <font color="red">10</font>
+| || &middot; || || 1 || || 1 || || 36|| || 1 || || 2 || || 1 || || 8 || || &middot; ||
+| <font color="red">10</font>
+|-
+| || || || ||
+| <font color="red">11</font>
+| || &middot; || || 1 || || 1 || || 49 || || 1 || || 5 || || 1 || || 27 || || 1 || || &middot; ||
+| <font color="red">11</font>
+|-
+| || || ||
+| <font color="red">12</font>
+| || &middot; || || 10 || || 3 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || &middot; ||
+| <font color="red">12</font>
+|-
+| || ||
+| <font color="red">13</font>
+| || &middot; || || 11 || || 1 || || 1 || || 2 || || 7 || || 1 || || 125 || || 4 || || 3 || || 1 || || &middot; ||
+| <font color="red">13</font>
+|-
+| ||
+| <font color="red">14</font>
+| || &middot; || || 3 || || 1 || || 100 || || 1 || || 1 || || 1 || || 216 || || 1 || || 1 || || 1 || || 4 || || &middot; ||
+| <font color="red">14</font>
+|-
+|
+| <font color="red">15</font>
+| || &middot; || || 13 || || 2 || || 121 || || 1 || || 3 || || 1 || || 343 || || 1 || || 5 || || 1 || || 9 || || 1 || || &middot; ||
+| <font color="red">15</font>
+|-
+| width="3%" | <font color="red">16</font>
+| width="3%" |
+| width="3%" | &middot;
+| width="3%" |
+| width="3%" | 14
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 9
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 10
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 6
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 2
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 2
+| width="3%" |
+| width="3%" | &middot;
+| width="3%" |
+| width="3%" | <font color="red">16</font>
+|}
+===TeX Smallmatrix===
+{| align="center"
+|
+<math>\begin{smallmatrix}
+&&&&&&&&&&&&&&& {\color{red}1} && {\color{red}1}
+\\
+&&&&&&&&&&&&&& {\color{red}2} && \cdot & & {\color{red}2}
+\\
+&&&&&&&&&&&&& {\color{red}3} && \cdot && \cdot && {\color{red}3}
+\\
+&&&&&&&&&&&& {\color{red}4} && \cdot && 2 && \cdot && {\color{red}4}
+\\
+&&&&&&&&&&& {\color{red}5} && \cdot && 3 && 1 && \cdot && {\color{red}5}
+\\
+&&&&&&&&&& {\color{red}6} && \cdot && 1 && 1 && 4 && \cdot && {\color{red}6}
+\\
+&&&&&&&&& {\color{red}7} && \cdot && 5 && 2 && 9 && 1 && \cdot && {\color{red}7}
+\\
+&&&&&&&& {\color{red}8} && \cdot && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}8}
+\\
+&&&&&&& {\color{red}9} && \cdot && 7 && 1 && 25 && 1 && 3 && 1 && \cdot && {\color{red}9}
+\\
+&&&&&& {\color{red}10} && \cdot && 1 && 1 && 36 && 1 && 2 && 1 && 8 && \cdot && {\color{red}10}
+\\
+&&&&& {\color{red}11} && \cdot && 1 && 1 && 49 && 1 && 5 && 1 && 27 && 1 && \cdot && {\color{red}11}
+\\
+&&&& {\color{red}12} && \cdot && 10 && 3 && 1 && 1 && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}12}
+\\
+&&& {\color{red}13} && \cdot && 11 && 1 && 1 && 2 && 7 && 1 && 125 && 4 && 3 && 1 && \cdot && {\color{red}13}
+\\
+&& {\color{red}14} && \cdot && 3 && 1 && 100 && 1 && 1 && 1 && 216 && 1 && 1 && 1 && 4 && \cdot && {\color{red}14}
+\\
+& {\color{red}15} && \cdot && 13 && 2 && 121 && 1 && 3 && 1 && 343 && 1 && 5 && 1 && 9 && 1 && \cdot && {\color{red}15}
+\\
+{\color{red}16} && \cdot && 14 && 1 && 9 && 1 && 10 && 1 && 1 && 1 && 6 && 1 && 2 && 1 && 2 && \cdot && {\color{red}16}
+\end{smallmatrix}</math>
+|}
+===ASCII===
+<pre>
+ Example
+    *                                   n o m
+    *                                    \ /
+    *                                   1 . 1
+    *                                  \ / \ /
+    *                                 2 .   . 2
+    *                                \ / \ / \ /
+    *                               3 .   .   . 3
+    *                              \ / \ / \ / \ /
+    *                             4 .   . 2 .   . 4
+    *                            \ / \ / \ / \ / \ /
+    *                           5 .   . 3 . 1 .   . 5
+    *                          \ / \ / \ / \ / \ / \ /
+    *                         6 .   . 1 . 1 . 4 .   . 6
+    *                        \ / \ / \ / \ / \ / \ / \ /
+    *                       7 .   . 5 . 2 . 9 . 1 .   . 7
+    *                      \ / \ / \ / \ / \ / \ / \ / \ /
+    *                     8 .   . 6 . 1 . 1 . 1 . 2 .   . 8
+    *                    \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *                   9 .   . 7 . 1 . 25. 1 . 3 . 1 .   . 9
+    *                  \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *                10 .   . 1 . 1 . 36. 1 . 2 . 1 . 8 .   . 10
+    *                \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *              11 .   . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 .   . 11
+    *              \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *            12 .   . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 .   . 12
+    *            \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *          13 .   . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 .   . 13
+    *          \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *        14 .   . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 .   . 14
+    *        \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *      15 .   . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 .   . 15
+    *      \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *    16 .   . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 .   . 16
+</pre>
+==A108352==
+* [http://oeis.org/wiki/A108352 A108352]
+===Links===
+* Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002846.html Primal Code Characteristic, n = 1 to 1000]
+* Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002847.html Primal Code Characteristic, n = 1001 to 2000]
+* Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002853.html Primal Code Characteristic, n = 2001 to 3000]
+===TeX Array===
+{| align="center"
+|
+<math>\begin{array}{*{10}{l}}
+a(1)
+& = & 1
+& \text{because} & (\circ~ 1)^1
+& = & (\circ~ \varnothing)^1
+& = & 1.
+\\
+a(2)
+& = & 0
+& \text{because} & (\circ~ 2)^k
+& = & (\circ~ 1\!:\!1)^k
+& = & 2,
+& \text{for all}~ k > 0.
+\\
+a(3)
+& = & 2
+& \text{because} & (\circ~ 3)^2
+& = & (\circ~ 2\!:\!1)^2
+& = & 1.
+\\
+a(4)
+& = & 2
+& \text{because} & (\circ~ 4 )^2
+& = & (\circ~ 1\!:\!2)^2
+& = &1.
+\\
+a(5)
+& = & 2
+& \text{because} & (\circ~ 5)^2
+& = & (\circ~ 3\!:\!1)^2
+& = & 1.
+\\
+a(6)
+& = & 0
+& \text{because} & (\circ~ 6)^k
+& = & (\circ~ 1\!:\!1 ~~ 2\!:\!1)^k
+& = & 6,
+& \text{for all}~ k > 0.
+\\
+a(7)
+& = & 2
+& \text{because} & (\circ~ 7)^2
+& = & (\circ~ 4\!:\!1)^1
+& = & 1.
+\\
+a(8)
+& = & 2
+& \text{because} & (\circ~ 8)^2
+& = & (\circ~ 1\!:\!3)^1
+& = & 1.
+\\
+a(9)
+& = & 0
+& \text{because} & (\circ~ 9)^k
+& = & (\circ~ 2\!:\!2)^k
+& = & 9,
+& \text{for all}~ k > 0.
+\\
+a(10)
+& = & 0
+& \text{because} & (\circ~ 10)^k
+& = & (\circ~ 1\!:\!1 ~~ 3\!:\!1)^k
+& = & 10,
+& \text{for all}~ k > 0.
+\end{array}</math>
+|}
+===ASCII===
+<pre>
+Example
+    * a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1.
+    * a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k.
+    * a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1.
+    * a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1.
+    * a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1.
+    * a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
+    * a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1.
+    * a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1.
+    * a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k.
+    * a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.
+    * Detail of calculation for compositional powers of 12:
+    * (12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
+    * (12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12
+    * Detail of calculation for compositional powers of 20:
+    * (20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
+    * (20 o)^3 = (3:2) o (1:2 3:1) = 1
+</pre>
+==A108371==
+* [http://oeis.org/wiki/A108371 A108371]
+===Wiki Table===
+{| align="center" style="font-weight:bold; text-align:center; width:90%"
+| || || || || || || || || || || || || || ||
+| <font color="red">1</font>
+|
+| <font color="red">1</font>
+|-
+| || || || || || || || || || || || || ||
+| <font color="red">2</font>
+| || 1 ||
+| <font color="red">2</font>
+|-
+| || || || || || || || || || || || ||
+| <font color="red">3</font>
+| || 2 || || 1 ||
+| <font color="red">3</font>
+|-
+| || || || || || || || || || || ||
+| <font color="red">4</font>
+| || 3 || || 2 || || 1 ||
+| <font color="red">4</font>
+|-
+| || || || || || || || || || ||
+| <font color="red">5</font>
+| || 4 || || 1 || || 2 || || 1 ||
+| <font color="red">5</font>
+|-
+| || || || || || || || || ||
+| <font color="red">6</font>
+| || 5 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">6</font>
+|-
+| || || || || || || || ||
+| <font color="red">7</font>
+| || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">7</font>
+|-
+| || || || || || || ||
+| <font color="red">8</font>
+| || 7 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">8</font>
+|-
+| || || || || || ||
+| <font color="red">9</font>
+| || 8 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">9</font>
+|-
+| || || || || ||
+| <font color="red">10</font>
+| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">10</font>
+|-
+| || || || ||
+| <font color="red">11</font>
+| || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">11</font>
+|-
+| || || ||
+| <font color="red">12</font>
+| || 11|| || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">12</font>
+|-
+| || ||
+| <font color="red">13</font>
+| || 12|| || 1 || || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">13</font>
+|-
+| ||
+| <font color="red">14</font>
+| || 13|| || 18|| || 1 || || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">14</font>
+|-
+|
+| <font color="red">15</font>
+| || 14 || || 1 || || 12 || || 1 || || 10 || || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 ||
+| <font color="red">15</font>
+|-
+| width="3%" | <font color="red">16</font>
+| width="3%" |
+| width="3%" | 15
+| width="3%" |
+| width="3%" | 14
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 18
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 10
+| width="3%" |
+| width="3%" | 9
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 6
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | 2
+| width="3%" |
+| width="3%" | 1
+| width="3%" |
+| width="3%" | <font color="red">16</font>
+|}
+===ASCII===
+<pre>
+ Example
+    * Table: T(n,k) = (n o)^k
+    *                                  T(n,k)
+    *                                    \ /
+    *                                   1 . 1
+    *                                  \ / \ /
+    *                                 2 . 1 . 2
+    *                                \ / \ / \ /
+    *                               3 . 2 . 1 . 3
+    *                              \ / \ / \ / \ /
+    *                             4 . 3 . 2 . 1 . 4
+    *                            \ / \ / \ / \ / \ /
+    *                           5 . 4 . 1 . 2 . 1 . 5
+    *                          \ / \ / \ / \ / \ / \ /
+    *                         6 . 5 . 1 . 1 . 2 . 1 . 6
+    *                        \ / \ / \ / \ / \ / \ / \ /
+    *                       7 . 6 . 1 . 1 . 1 . 2 . 1 . 7
+    *                      \ / \ / \ / \ / \ / \ / \ / \ /
+    *                     8 . 7 . 6 . 1 . 1 . 1 . 2 . 1 . 8
+    *                    \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *                   9 . 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 9
+    *                  \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *                10 . 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 10
+    *                \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *              11 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 11
+    *              \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *            12 . 11. 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 12
+    *            \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *          13 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 13
+    *          \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *        14 . 13. 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 14
+    *        \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *      15 . 14. 1 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 15
+    *      \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
+    *    16 . 15. 14. 1 . 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 16
+</pre>
+==A109300==
+* [http://oeis.org/wiki/A109300 A109300]
+===JPEG===
+{| align="center" border="1" cellpadding="10"
+|
+<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
+<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
+|
+<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
+<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
+|
+<p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
+|
+<p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
+<p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p>
+|
+<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p>
+|
+<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
+<p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p>
+|
+<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
+<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p>
+|}
+===ASCII===
+<pre>
+ Example
+    * Table of Rotes and Primal Functions for Positive Integers of Rote Height 2
+    *
+    * o-o     o-o       o-o   o-o o-o     o-o o-o       o-o o-o     o-o o-o o-o
+    * |       |         |     |   |       |   |         |   |       |   |   |
+    * o-o   o-o     o-o o-o   o---o     o-o   o-o   o-o o---o     o-o   o---o
+    * |     |       |   |     |         |     |     |   |         |     |
+    * O     O       O===O     O         O=====O     O===O         O=====O
+    *
+    * 2:1   1:2     1:1 2:1   2:2       1:2 2:1     1:1 2:2       1:2 2:2
+    *
+    * 3     4       6         9         12          18            36
+    *
+</pre>
+==A109301==
+* [http://oeis.org/wiki/A109301 A109301]
+===Example===
+: <math>802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1</math>
+: <math>\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}</math>
+: <math>\begin{array}{lllll}
+& = & 9 \cdot 89189
+& = & 2\!:\!2 ~~ 8638\!:\!1
+\\
+& = & 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>\text{p}\!</math></p><br>
+<p><math>a(2) ~=~ 1</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 3 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_\text{p}\!</math></p><br>
+<p><math>a(3) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 4 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^\text{p}\!</math></p><br>
+<p><math>a(4) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 5 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(5) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}\!</math></p><br>
+<p><math>a(6) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 7 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(7) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 8 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(8) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 9 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>a(9) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 10 Big.jpg|80px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(10) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 11 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(11) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br>
+<p><math>a(12) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 13 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(13) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 14 Big.jpg|105px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(14) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 15 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(15) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 16 Big.jpg|90px]]</p><br>
+<p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(16) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 17 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(17) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 18 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>a(18) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 19 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(19) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 20 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(20) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 21 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(21) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 22 Big.jpg|80px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(22) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 23 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(23) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 24 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br>
+<p><math>a(24) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 25 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
+<p><math>a(25) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 26 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(26) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 27 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(27) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 28 Big.jpg|130px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(28) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 29 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(29) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 30 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(30) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 31 Big.jpg|40px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br>
+<p><math>a(31) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 32 Big.jpg|65px]]</p><br>
+<p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(32) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 33 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(33) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 34 Big.jpg|105px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(34) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 35 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(35) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 36 Big.jpg|145px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
+<p><math>a(36) ~=~ 2</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 37 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(37) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 38 Big.jpg|105px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(38) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 39 Big.jpg|120px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(39) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 40 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(40) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 41 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(41) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 42 Big.jpg|145px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(42) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 43 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(43) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 44 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(44) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 45 Big.jpg|120px]]</p><br>
+<p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(45) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 46 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(46) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 47 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(47) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 48 Big.jpg|105px]]</p><br>
+<p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br>
+<p><math>a(48) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 49 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br>
+<p><math>a(49) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 50 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br>
+<p><math>a(50) ~=~ 3</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 51 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(51) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 52 Big.jpg|145px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br>
+<p><math>a(52) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 53 Big.jpg|90px]]</p><br>
+<p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br>
+<p><math>a(53) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 54 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(54) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 55 Big.jpg|80px]]</p><br>
+<p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(55) ~=~ 4</math></p>
+|-
+| valign="bottom" |
+<p>[[Image:Rote 56 Big.jpg|130px]]</p><br>
+<p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br>
+<p><math>a(56) ~=~ 3</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 57 Big.jpg|105px]]</p><br>
+<p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(57) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 58 Big.jpg|120px]]</p><br>
+<p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br>
+<p><math>a(58) ~=~ 4</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 59 Big.jpg|65px]]</p><br>
+<p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br>
+<p><math>a(59) ~=~ 5</math></p>
+| valign="bottom" |
+<p>[[Image:Rote 60 Big.jpg|155px]]</p><br>
+<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
+<p><math>a(60) ~=~ 3</math></p>
+|}
+===ASCII===
+<pre>
+  Comment
+     * Table of Rotes and Primal Functions for Positive Integers from 1 to 40
+     *
+     *                                                         o-o
+     *                                                         |
+     *                             o-o             o-o         o-o
+     *                             |               |           |
+     *               o-o           o-o           o-o           o-o
+     *               |             |             |             |
+     * O             O             O             O             O
+     *
+     * { }           1:1           2:1           1:2           3:1
+     *
+     * 1             2             3             4             5
+     *
+     *
+     *                 o-o           o-o                           o-o
+     *                 |             |                             |
+     *     o-o       o-o             o-o         o-o o-o           o-o
+     *     |         |               |           |   |             |
+     * o-o o-o       o-o           o-o           o---o         o-o o-o
+     * |   |         |             |             |             |   |
+     * O===O         O             O             O             O===O
+     *
+     * 1:1 2:1       4:1           1:3           2:2           1:1 3:1
+     *
+     * 6             7             8             9             10
+     *
+     *
+     * o-o
+     * |
+     * o-o                             o-o             o-o         o-o
+     * |                               |               |           |
+     * o-o             o-o o-o     o-o o-o           o-o       o-o o-o
+     * |               |   |       |   |             |         |   |
+     * o-o           o-o   o-o     o===o-o       o-o o-o       o-o o-o
+     * |             |     |       |             |   |         |   |
+     * O             O=====O       O             O===O         O===O
+     *
+     * 5:1           1:2 2:1       6:1           1:1 4:1       2:1 3:1
+     *
+     * 11            12            13            14            15
+     *
+     *
+     *                 o-o                         o-o
+     *                 |                           |
+     *     o-o       o-o                           o-o               o-o
+     *     |         |                             |                 |
+     *   o-o         o-o               o-o o-o   o-o             o-o o-o
+     *   |           |                 |   |     |               |   |
+     * o-o           o-o           o-o o---o     o-o           o-o   o-o
+     * |             |             |   |         |             |     |
+     * O             O             O===O         O             O=====O
+     *
+     * 1:4           7:1           1:1 2:2       8:1           1:2 3:1
+     *
+     * 16            17            18            19            20
+     *
+     *
+     *                   o-o
+     *                   |
+     *       o-o         o-o       o-o o-o         o-o         o-o
+     *       |           |         |   |           |           |
+     * o-o o-o           o-o       o---o           o-o o-o     o-o o-o
+     * |   |             |         |               |   |       |   |
+     * o-o o-o       o-o o-o       o-o           o-o   o-o     o---o
+     * |   |         |   |         |             |     |       |
+     * O===O         O===O         O             O=====O       O
+     *
+     * 2:1 4:1       1:1 5:1       9:1           1:3 2:1       3:2
+     *
+     * 21            22            23            24            25
+     *
+     *
+     *                                               o-o
+     *                                               |
+     *         o-o       o-o               o-o       o-o               o-o
+     *         |         |                 |         |                 |
+     *     o-o o-o   o-o o-o         o-o o-o     o-o o-o           o-o o-o
+     *     |   |     |   |           |   |       |   |             |   |
+     * o-o o===o-o   o---o         o-o   o-o     o===o-o       o-o o-o o-o
+     * |   |         |             |     |       |             |   |   |
+     * O===O         O             O=====O       O             O===O===O
+     *
+     * 1:1 6:1       2:3           1:2 4:1       10:1          1:1 2:1 3:1
+     *
+     * 26            27            28            29            30
+     *
+     *
+     * o-o
+     * |
+     * o-o             o-o             o-o             o-o
+     * |               |               |               |
+     * o-o             o-o             o-o           o-o       o-o   o-o
+     * |               |               |             |         |     |
+     * o-o             o-o         o-o o-o           o-o       o-o o-o
+     * |               |           |   |             |         |   |
+     * o-o           o-o           o-o o-o       o-o o-o       o-o o-o
+     * |             |             |   |         |   |         |   |
+     * O             O             O===O         O===O         O===O
+     *
+     * 11:1          1:5           2:1 5:1       1:1 7:1       3:1 4:1
+     *
+     * 31            32            33            34            35
+     *
+     *
+     *                                   o-o
+     *                                   |
+     *                 o-o o-o           o-o             o-o     o-o o-o
+     *                 |   |             |               |       |   |
+     *   o-o o-o o-o o-o   o-o         o-o       o-o o-o o-o     o-o o-o
+     *   |   |   |   |     |           |         |   |   |       |   |
+     * o-o   o---o   o=====o-o     o-o o-o       o-o o===o-o   o-o   o-o
+     * |     |       |             |   |         |   |         |     |
+     * O=====O       O             O===O         O===O         O=====O
+     *
      * 1:2 2:2       12:1          1:1 8:1       2:1 6:1       1:3 3:1
      *
      * 36            37            38            39            40
      *
      * In these Figures, "extended lines of identity" like o===o
      * indicate identified nodes and capital O is the root node.
      * The rote height in gammas is found by finding the number
      * of graphs of the following shape between the root and one
      * of the highest nodes of the tree:
      * o--o
      * |
      * o
      * A sequence like this, that can be regarded as a nonnegative integer
      * measure on positive integers, may have as many as 3 other sequences
      * associated with it. Given that the fiber of a function f at n is all
      * the domain elements that map to n, we always have the fiber minimum
      * or minimum inverse function and may also have the fiber cardinality
      * and the fiber maximum or maximum inverse function. For A109301, the
      * minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the
      * first positive integer whose rote height is n, the fiber cardinality
      * is A109300, giving the number of positive integers of rote height n,
      * while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving
      * the last positive integer whose rote height is n, has the following
      * initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36,
      * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
      * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
      * 7.840858554516122655953405327738 x 10^371.
+ Example
+    * Writing (prime(i))^j as i:j, we have:
+    * 802701 = 2:2 8638:1
+    * 8638 = 1:1 4:1 113:1
+    * 113 = 30:1
+    * 30 = 1:1 2:1 3:1
+    * 4 = 1:2
+    * 3 = 2:1
+    * 2 = 1:1
+    * 1 = { }
+    * So rote(802701) is the graph:
+    *
+    *                           o-o
+    *                           |
+    *                       o-o o-o
+    *                       |   |
+    *               o-o o-o o-o o-o
+    *               |   |   |   |
+    *             o-o   o===o===o-o
+    *             |     |
+    * o-o o-o o-o o-o   o---------o
+    * |   |   |   |     |
+    * o---o   o===o=====o---------o
+    * |       |
+    * O=======O
+    *
+    * Therefore rhig(802701) = 6.
 </pre>
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