Difference between revisions of "Directory:Jon Awbrey/Papers/Riffs and Rotes"

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==Idea==
 
==Idea==
  
Let <math>\text{p}_i\!</math> be the <math>i^\text{th}\!</math> prime, taking <math>2\!</math> as <math>1^\text{st}\!</math> prime in the sequence that begins as follows:
+
Let <math>\text{p}_i\!</math> be the <math>i^\text{th}\!</math> prime, taking <math>2\!</math> as the <math>1^\text{st}\!</math> prime in the sequence that begins as follows:
  
: <math>\text{p}_1 = 2, \text{p}_2 = 3, \text{p}_3 = 5, \text{p}_4 = 7, \text{p}_5 = 11\!</math> and so on.
+
{| align="center" cellpadding="6" width="90%"
 
+
|
{| align="center" cellpadding="20" width="100%"
+
<math>\begin{matrix}
| <math>\cdots</math>
+
\text{p}_1 = 2, &
 +
\text{p}_2 = 3, &
 +
\text{p}_3 = 5, &
 +
\text{p}_4 = 7, &
 +
\text{p}_5 = 11, &
 +
\text{p}_6 = 13, &
 +
\ldots
 +
\end{matrix}</math>
 
|}
 
|}
  

Revision as of 03:14, 3 February 2010

Idea

Let \(\text{p}_i\!\) be the \(i^\text{th}\!\) prime, taking \(2\!\) as the \(1^\text{st}\!\) prime in the sequence that begins as follows:

\(\begin{matrix} \text{p}_1 = 2, & \text{p}_2 = 3, & \text{p}_3 = 5, & \text{p}_4 = 7, & \text{p}_5 = 11, & \text{p}_6 = 13, & \ldots \end{matrix}\)

Riffs in Numerical Order

\(\text{Riffs in Numerical Order}\!\)

 


\(1\!\)


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

Riff 2 Big.jpg


\(\text{p}\!\)


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

Riff 3 Big.jpg


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


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

Riff 4 Big.jpg


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


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

Riff 5 Big.jpg


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


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

Riff 6 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)

Riff 7 Big.jpg


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


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

Riff 8 Big.jpg


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


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

Riff 9 Big.jpg


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


\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)

Riff 10 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)

Riff 11 Big.jpg


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


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

Riff 12 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)

Riff 13 Big.jpg


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


\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)

Riff 14 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)

Riff 15 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)

Riff 16 Big.jpg


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


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

Riff 17 Big.jpg


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


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

Riff 18 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)

Riff 19 Big.jpg


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


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

Riff 20 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)

Riff 21 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)

Riff 22 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)

Riff 23 Big.jpg


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


\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)

Riff 24 Big.jpg


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


\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)

Riff 25 Big.jpg


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


\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)

Riff 26 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)

Riff 27 Big.jpg


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


\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)

Riff 28 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)

Riff 29 Big.jpg


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


\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)

Riff 30 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)

Riff 31 Big.jpg


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


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

Riff 32 Big.jpg


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


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

Riff 33 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)

Riff 34 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)

Riff 35 Big.jpg


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


\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)

Riff 36 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)

Riff 37 Big.jpg


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


\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)

Riff 38 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)

Riff 39 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)

Riff 40 Big.jpg


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


\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)

Riff 41 Big.jpg


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


\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)

Riff 42 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)

Riff 43 Big.jpg


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


\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)

Riff 44 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)

Riff 45 Big.jpg


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


\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)

Riff 46 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)

Riff 47 Big.jpg


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


\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)

Riff 48 Big.jpg


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


\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)

Riff 49 Big.jpg


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


\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)

Riff 50 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)

Riff 51 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)

Riff 52 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)

Riff 53 Big.jpg


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


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

Riff 54 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)

Riff 55 Big.jpg


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


\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)

Riff 56 Big.jpg


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


\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)

Riff 57 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)

Riff 58 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)

Riff 59 Big.jpg


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


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

Riff 60 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Rotes in Numerical Order

Rote 1 Big.jpg


\(1\!\)


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

Rote 2 Big.jpg


\(\text{p}\!\)


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

Rote 3 Big.jpg


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


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

Rote 4 Big.jpg


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


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

Rote 5 Big.jpg


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


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

Rote 6 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)

Rote 7 Big.jpg


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


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

Rote 8 Big.jpg


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


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

Rote 9 Big.jpg


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


\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)

Rote 10 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)

Rote 11 Big.jpg


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


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

Rote 12 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)

Rote 13 Big.jpg


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


\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)

Rote 14 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)

Rote 15 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)

Rote 16 Big.jpg


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


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

Rote 17 Big.jpg


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


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

Rote 18 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)

Rote 19 Big.jpg


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


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

Rote 20 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)

Rote 21 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)

Rote 22 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)

Rote 23 Big.jpg


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


\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)

Rote 24 Big.jpg


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


\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)

Rote 25 Big.jpg


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


\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)

Rote 26 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)

Rote 27 Big.jpg


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


\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)

Rote 28 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)

Rote 29 Big.jpg


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


\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)

Rote 30 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)

Rote 31 Big.jpg


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


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

Rote 32 Big.jpg


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


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

Rote 33 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)

Rote 34 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)

Rote 35 Big.jpg


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


\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)

Rote 36 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)

Rote 37 Big.jpg


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


\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)

Rote 38 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)

Rote 39 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)

Rote 40 Big.jpg


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


\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)

Rote 41 Big.jpg


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


\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)

Rote 42 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)

Rote 43 Big.jpg


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


\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)

Rote 44 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)

Rote 45 Big.jpg


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


\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)

Rote 46 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)

Rote 47 Big.jpg


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


\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)

Rote 48 Big.jpg


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


\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)

Rote 49 Big.jpg


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


\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)

Rote 50 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)

Rote 51 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)

Rote 52 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)

Rote 53 Big.jpg


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


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

Rote 54 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)

Rote 55 Big.jpg


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


\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)

Rote 56 Big.jpg


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


\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)

Rote 57 Big.jpg


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


\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)

Rote 58 Big.jpg


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


\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)

Rote 59 Big.jpg


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


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

Rote 60 Big.jpg


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


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Selected Sequences

A061396

  • Number of "rooted index-functional forests" (Riffs) on n nodes.
  • Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.
\(\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!\)
\(\text{Integer}\!\) \(\text{Factorization}\!\) \(\text{Notation}\!\) \(\text{Riff Digraph}\!\) \(\text{Rote Graph}\!\) \(\text{Traversal}\!\)
\(1\!\) \(1\!\)     Rote 1 Big.jpg  
\(2\!\) \(\text{p}_1^1\!\) \(\text{p}\!\) Riff 2 Big.jpg Rote 2 Big.jpg \(((~))\)
\(3\!\)

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

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

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

\(\text{p}^\text{p}\!\) Riff 4 Big.jpg Rote 4 Big.jpg \(((((~))))\)
\(5\!\)

\(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \'"`UNIQ-MathJax1-QINU`"' '"`UNIQ-MathJax2-QINU`"' '"`UNIQ-MathJax3-QINU`"' '"`UNIQ-MathJax4-QINU`"' :{| border="1" cellpadding="20" | [[Image:Rote 802701 Big.jpg|330px]] |} '"`UNIQ-MathJax5-QINU`"' <br> {| align="center" border="1" cellpadding="6" |+ style="height:25px" | \(a(n) = \text{Rote Height of}~ n\)

Rote 1 Big.jpg


\(1\!\)


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

Rote 2 Big.jpg


\(\text{p}\!\)


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

Rote 3 Big.jpg


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


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

Rote 4 Big.jpg


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


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

Rote 5 Big.jpg


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


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

Rote 6 Big.jpg


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


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

Rote 7 Big.jpg


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


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

Rote 8 Big.jpg


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


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

Rote 9 Big.jpg


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


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

Rote 10 Big.jpg


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


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

Rote 11 Big.jpg


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


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

Rote 12 Big.jpg


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


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

Rote 13 Big.jpg


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


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

Rote 14 Big.jpg


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


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

Rote 15 Big.jpg


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


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

Rote 16 Big.jpg


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


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

Rote 17 Big.jpg


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


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

Rote 18 Big.jpg


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


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

Rote 19 Big.jpg


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


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

Rote 20 Big.jpg


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


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

Rote 21 Big.jpg


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


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

Rote 22 Big.jpg


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


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

Rote 23 Big.jpg


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


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

Rote 24 Big.jpg


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


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

Rote 25 Big.jpg


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


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

Rote 26 Big.jpg


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


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

Rote 27 Big.jpg


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


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

Rote 28 Big.jpg


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


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

Rote 29 Big.jpg


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


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

Rote 30 Big.jpg


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


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

Rote 31 Big.jpg


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


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

Rote 32 Big.jpg


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


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

Rote 33 Big.jpg


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


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

Rote 34 Big.jpg


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


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

Rote 35 Big.jpg


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


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

Rote 36 Big.jpg


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


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

Rote 37 Big.jpg


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


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

Rote 38 Big.jpg


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


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

Rote 39 Big.jpg


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


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

Rote 40 Big.jpg


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


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

Rote 41 Big.jpg


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


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

Rote 42 Big.jpg


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


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

Rote 43 Big.jpg


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


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

Rote 44 Big.jpg


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


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

Rote 45 Big.jpg


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


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

Rote 46 Big.jpg


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


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

Rote 47 Big.jpg


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


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

Rote 48 Big.jpg


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


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

Rote 49 Big.jpg


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


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

Rote 50 Big.jpg


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


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

Rote 51 Big.jpg


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


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

Rote 52 Big.jpg


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


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

Rote 53 Big.jpg


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


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

Rote 54 Big.jpg


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


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

Rote 55 Big.jpg


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


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

Rote 56 Big.jpg


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


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

Rote 57 Big.jpg


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


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

Rote 58 Big.jpg


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


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

Rote 59 Big.jpg


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


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

Rote 60 Big.jpg


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


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