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

MyWikiBiz, Author Your Legacy — Friday November 08, 2024
Jump to navigationJump to search
Line 1,062: Line 1,062:
 
4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20
 
4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20
 
\end{array}</math>
 
\end{array}</math>
 +
|}
 +
 +
{| align="center" border="1" width="90%"
 +
|+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math>
 +
|- style="height:50px; background:#f0f0ff"
 +
|
 +
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
 +
| 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>
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>1\!</math>
 +
| width="25%" | <math>1\!</math>
 +
| width="15%" | &nbsp;
 +
| width="25%" | &nbsp;
 +
| width="25%" | [[Image:Rote 1 Big.jpg|20px]]
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| 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]]
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| 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>
 +
|
 +
<math>\begin{array}{lll}
 +
\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]]
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>5\!</math>
 +
| width="25%" |
 +
<math>\begin{array}{lll}
 +
\text{p}_3^1
 +
& = & \text{p}_{\text{p}_2^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
 +
\end{array}</math>
 +
| width="15%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
 +
| width="25%" | [[Image:Riff 5 Big.jpg|65px]]
 +
| width="25%" | [[Image:Rote 5 Big.jpg|40px]]
 +
|-
 +
| <math>6\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^1 \text{p}_2^1
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
 +
\end{array}</math>
 +
| <math>\text{p} \text{p}_{\text{p}}\!</math>
 +
| [[Image:Riff 6 Big.jpg|65px]]
 +
| [[Image:Rote 6 Big.jpg|80px]]
 +
|-
 +
| <math>7\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_4^1
 +
& = & \text{p}_{\text{p}_1^2}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
 +
| [[Image:Riff 7 Big.jpg|65px]]
 +
| [[Image:Rote 7 Big.jpg|65px]]
 +
|-
 +
| <math>8\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^3
 +
& = & \text{p}_1^{\text{p}_2^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 8 Big.jpg|65px]]
 +
| [[Image:Rote 8 Big.jpg|65px]]
 +
|-
 +
| <math>9\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_2^2
 +
& = & \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]]
 +
|-
 +
| <math>16\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^4
 +
& = & \text{p}_1^{\text{p}_1^2}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
 +
| [[Image:Riff 16 Big.jpg|65px]]
 +
| [[Image:Rote 16 Big.jpg|90px]]
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>10\!</math>
 +
| width="25%" |
 +
<math>\begin{array}{lll}
 +
\text{p}_1^1 \text{p}_3^1
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1
 +
\\[12pt]
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
 +
\end{array}</math>
 +
| width="15%" | <math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math>
 +
| width="25%" | [[Image:Riff 10 Big.jpg|90px]]
 +
| width="25%" | [[Image:Rote 10 Big.jpg|80px]]
 +
|-
 +
| <math>11\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_5^1
 +
& = & \text{p}_{\text{p}_3^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math>
 +
| [[Image:Riff 11 Big.jpg|90px]]
 +
| [[Image:Rote 11 Big.jpg|40px]]
 +
|-
 +
| <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]]
 +
|-
 +
| <math>13\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_6^1
 +
& = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 13 Big.jpg|65px]]
 +
| [[Image:Rote 13 Big.jpg|80px]]
 +
|-
 +
| <math>14\!</math>
 +
|
 +
<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]]
 +
|-
 +
| <math>17\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_7^1
 +
& = & \text{p}_{\text{p}_4^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 17 Big.jpg|90px]]
 +
| [[Image:Rote 17 Big.jpg|65px]]
 +
|-
 +
| <math>18\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^1 \text{p}_2^2
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math>
 +
| [[Image:Riff 18 Big.jpg|65px]]
 +
| [[Image:Rote 18 Big.jpg|120px]]
 +
|-
 +
| <math>19\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_8^1
 +
& = & \text{p}_{\text{p}_1^3}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math>
 +
| [[Image:Riff 19 Big.jpg|90px]]
 +
| [[Image:Rote 19 Big.jpg|65px]]
 +
|-
 +
| <math>23\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_9^1
 +
& = & \text{p}_{\text{p}_2^2}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math>
 +
| [[Image:Riff 23 Big.jpg|65px]]
 +
| [[Image:Rote 23 Big.jpg|80px]]
 +
|-
 +
| <math>25\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_3^2
 +
& = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math>
 +
| [[Image:Riff 25 Big.jpg|65px]]
 +
| [[Image:Rote 25 Big.jpg|80px]]
 +
|-
 +
| <math>27\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_2^3
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 27 Big.jpg|65px]]
 +
| [[Image:Rote 27 Big.jpg|80px]]
 +
|-
 +
| <math>32\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^5
 +
& = & \text{p}_1^{\text{p}_3^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math>
 +
| [[Image:Riff 32 Big.jpg|90px]]
 +
| [[Image:Rote 32 Big.jpg|65px]]
 +
|-
 +
| <math>49\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_4^2
 +
& = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math>
 +
| [[Image:Riff 49 Big.jpg|65px]]
 +
| [[Image:Rote 49 Big.jpg|80px]]
 +
|-
 +
| <math>53\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_{16}^1
 +
& = & \text{p}_{\text{p}_1^4}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 53 Big.jpg|90px]]
 +
| [[Image:Rote 53 Big.jpg|90px]]
 +
|-
 +
| <math>64\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^6
 +
& = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 64 Big.jpg|65px]]
 +
| [[Image:Rote 64 Big.jpg|105px]]
 +
|-
 +
| <math>81\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_2^4
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
 +
| [[Image:Riff 81 Big.jpg|65px]]
 +
| [[Image:Rote 81 Big.jpg|105px]]
 +
|-
 +
| <math>128\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^7
 +
& = & \text{p}_1^{\text{p}_4^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 128 Big.jpg|90px]]
 +
| [[Image:Rote 128 Big.jpg|90px]]
 +
|-
 +
| <math>256\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^8
 +
& = & \text{p}_1^{\text{p}_1^3}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math>
 +
| [[Image:Riff 256 Big.jpg|90px]]
 +
| [[Image:Rote 256 Big.jpg|90px]]
 +
|-
 +
| <math>512\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^9
 +
& = & \text{p}_1^{\text{p}_2^2}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math>
 +
| [[Image:Riff 512 Big.jpg|65px]]
 +
| [[Image:Rote 512 Big.jpg|105px]]
 +
|-
 +
| <math>65536\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^{16}
 +
& = & \text{p}_1^{\text{p}_1^4}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 65536 Big.jpg|90px]]
 +
| [[Image:Rote 65536 Big.jpg|115px]]
 +
|}
 
|}
 
|}
  

Revision as of 15:36, 25 January 2010

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

\(\text{Prime Factorizations, Riffs, and Rotes}\!\)
\(\text{Integer}\!\) \(\text{Factorization}\!\) \(\text{Notation}\!\) \(\text{Riff Digraph}\!\) \(\text{Rote Graph}\!\)
\(1\!\) \(1\!\)     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
\(5\!\)

\(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}}}\!\) Riff 5 Big.jpg Rote 5 Big.jpg
\(6\!\)

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

\(\text{p} \text{p}_{\text{p}}\!\) Riff 6 Big.jpg Rote 6 Big.jpg
\(7\!\)

\(\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}}}\!\) Riff 7 Big.jpg Rote 7 Big.jpg
\(8\!\)

\(\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}}}\!\) Riff 8 Big.jpg Rote 8 Big.jpg
\(9\!\)

\(\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\)

\(\text{p}_\text{p}^\text{p}\!\) Riff 9 Big.jpg Rote 9 Big.jpg
\(16\!\)

\(\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}\)

\(\text{p}^{\text{p}^{\text{p}}}\!\) Riff 16 Big.jpg Rote 16 Big.jpg
\(10\!\)

\(\begin{array}{lll} \text{p}_1^1 \text{p}_3^1 & = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\)

\(\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) Riff 10 Big.jpg Rote 10 Big.jpg
\(11\!\)

\(\begin{array}{lll} \text{p}_5^1 & = & \text{p}_{\text{p}_3^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) Riff 11 Big.jpg Rote 11 Big.jpg
\(12\!\)

\(\begin{array}{lll} \text{p}_1^2 \text{p}_2^1 & = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1 \end{array}\)

\(\text{p}^{\text{p}} \text{p}_{\text{p}}\!\) Riff 12 Big.jpg Rote 12 Big.jpg
\(13\!\)

\(\begin{array}{lll} \text{p}_6^1 & = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1 \\[12pt] & = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1 \end{array}\)

\(\text{p}_{\text{p} \text{p}_{\text{p}}}\!\) Riff 13 Big.jpg Rote 13 Big.jpg
\(14\!\)

\(\begin{array}{lll} \text{p}_1^1 \text{p}_4^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\)

\(\text{p} \text{p}_{\text{p}^{\text{p}}}\!\) Riff 14 Big.jpg Rote 14 Big.jpg
\(17\!\)

\(\begin{array}{lll} \text{p}_7^1 & = & \text{p}_{\text{p}_4^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!\) Riff 17 Big.jpg Rote 17 Big.jpg
\(18\!\)

\(\begin{array}{lll} \text{p}_1^1 \text{p}_2^2 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\)

\(\text{p} \text{p}_{\text{p}}^{\text{p}}\!\) Riff 18 Big.jpg Rote 18 Big.jpg
\(19\!\)

\(\begin{array}{lll} \text{p}_8^1 & = & \text{p}_{\text{p}_1^3}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!\) Riff 19 Big.jpg Rote 19 Big.jpg
\(23\!\)

\(\begin{array}{lll} \text{p}_9^1 & = & \text{p}_{\text{p}_2^2}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!\) Riff 23 Big.jpg Rote 23 Big.jpg
\(25\!\)

\(\begin{array}{lll} \text{p}_3^2 & = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1} \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1} \end{array}\)

\(\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!\) Riff 25 Big.jpg Rote 25 Big.jpg
\(27\!\)

\(\begin{array}{lll} \text{p}_2^3 & = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1} \\[12pt] & = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1} \end{array}\)

\(\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!\) Riff 27 Big.jpg Rote 27 Big.jpg
\(32\!\)

\(\begin{array}{lll} \text{p}_1^5 & = & \text{p}_1^{\text{p}_3^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!\) Riff 32 Big.jpg Rote 32 Big.jpg
\(49\!\)

\(\begin{array}{lll} \text{p}_4^2 & = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1} \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1} \end{array}\)

\(\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!\) Riff 49 Big.jpg Rote 49 Big.jpg
\(53\!\)

\(\begin{array}{lll} \text{p}_{16}^1 & = & \text{p}_{\text{p}_1^4}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!\) Riff 53 Big.jpg Rote 53 Big.jpg
\(64\!\)

\(\begin{array}{lll} \text{p}_1^6 & = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1} \\[12pt] & = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1} \end{array}\)

\(\text{p}^{\text{p} \text{p}_{\text{p}}}\!\) Riff 64 Big.jpg Rote 64 Big.jpg
\(81\!\)

\(\begin{array}{lll} \text{p}_2^4 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2} \\[12pt] & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}} \end{array}\)

\(\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!\) Riff 81 Big.jpg Rote 81 Big.jpg
\(128\!\)

\(\begin{array}{lll} \text{p}_1^7 & = & \text{p}_1^{\text{p}_4^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!\) Riff 128 Big.jpg Rote 128 Big.jpg
\(256\!\)

\(\begin{array}{lll} \text{p}_1^8 & = & \text{p}_1^{\text{p}_1^3} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}} \end{array}\)

\(\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!\) Riff 256 Big.jpg Rote 256 Big.jpg
\(512\!\)

\(\begin{array}{lll} \text{p}_1^9 & = & \text{p}_1^{\text{p}_2^2} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!\) Riff 512 Big.jpg Rote 512 Big.jpg
\(65536\!\)

\(\begin{array}{lll} \text{p}_1^{16} & = & \text{p}_1^{\text{p}_1^4} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}} \end{array}\)

\(\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!\) Riff 65536 Big.jpg Rote 65536 Big.jpg

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 \(((((~))))\)
\(5\!\)

\(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[10pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}}}\!\) Riff 5 Big.jpg Rote 5 Big.jpg \(((((~))(~))(~))\)
\(6\!\)

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

\(\text{p} \text{p}_{\text{p}}\!\) Riff 6 Big.jpg Rote 6 Big.jpg \(((~))(((~))(~))\)
\(7\!\)

\(\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[10pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}}}\!\) Riff 7 Big.jpg Rote 7 Big.jpg \((((((~))))(~))\)
\(8\!\)

\(\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[10pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}}}\!\) Riff 8 Big.jpg Rote 8 Big.jpg \((((((~))(~))))\)
\(9\!\)

\(\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\)

\(\text{p}_\text{p}^\text{p}\!\) Riff 9 Big.jpg Rote 9 Big.jpg \((((~))(((~))))\)
\(16\!\)

\(\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[10pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}\)

\(\text{p}^{\text{p}^{\text{p}}}\!\) Riff 16 Big.jpg Rote 16 Big.jpg \(((((((~))))))\)

A062504

  • Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.

\(\begin{array}{l|l|r} k & P_k = \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \} = \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \} & |P_k| \\[10pt] 0 & \{ 1 \} & 1 \\ 1 & \{ 2 \} & 1 \\ 2 & \{ 3, 4 \} & 2 \\ 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 \\ 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 \end{array}\)

\(\text{Prime Factorizations, Riffs, and Rotes}\!\)
\(\text{Integer}\!\) \(\text{Factorization}\!\) \(\text{Notation}\!\) \(\text{Riff Digraph}\!\) \(\text{Rote Graph}\!\)
\(1\!\) \(1\!\)     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
\(5\!\)

\(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}}}\!\) Riff 5 Big.jpg Rote 5 Big.jpg
\(6\!\)

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

\(\text{p} \text{p}_{\text{p}}\!\) Riff 6 Big.jpg Rote 6 Big.jpg
\(7\!\)

\(\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}}}\!\) Riff 7 Big.jpg Rote 7 Big.jpg
\(8\!\)

\(\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}}}\!\) Riff 8 Big.jpg Rote 8 Big.jpg
\(9\!\)

\(\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\)

\(\text{p}_\text{p}^\text{p}\!\) Riff 9 Big.jpg Rote 9 Big.jpg
\(16\!\)

\(\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}\)

\(\text{p}^{\text{p}^{\text{p}}}\!\) Riff 16 Big.jpg Rote 16 Big.jpg
\(10\!\)

\(\begin{array}{lll} \text{p}_1^1 \text{p}_3^1 & = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\)

\(\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) Riff 10 Big.jpg Rote 10 Big.jpg
\(11\!\)

\(\begin{array}{lll} \text{p}_5^1 & = & \text{p}_{\text{p}_3^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) Riff 11 Big.jpg Rote 11 Big.jpg
\(12\!\)

\(\begin{array}{lll} \text{p}_1^2 \text{p}_2^1 & = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1 \end{array}\)

\(\text{p}^{\text{p}} \text{p}_{\text{p}}\!\) Riff 12 Big.jpg Rote 12 Big.jpg
\(13\!\)

\(\begin{array}{lll} \text{p}_6^1 & = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1 \\[12pt] & = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1 \end{array}\)

\(\text{p}_{\text{p} \text{p}_{\text{p}}}\!\) Riff 13 Big.jpg Rote 13 Big.jpg
\(14\!\)

\(\begin{array}{lll} \text{p}_1^1 \text{p}_4^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\)

\(\text{p} \text{p}_{\text{p}^{\text{p}}}\!\) Riff 14 Big.jpg Rote 14 Big.jpg
\(17\!\)

\(\begin{array}{lll} \text{p}_7^1 & = & \text{p}_{\text{p}_4^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!\) Riff 17 Big.jpg Rote 17 Big.jpg
\(18\!\)

\(\begin{array}{lll} \text{p}_1^1 \text{p}_2^2 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\)

\(\text{p} \text{p}_{\text{p}}^{\text{p}}\!\) Riff 18 Big.jpg Rote 18 Big.jpg
\(19\!\)

\(\begin{array}{lll} \text{p}_8^1 & = & \text{p}_{\text{p}_1^3}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!\) Riff 19 Big.jpg Rote 19 Big.jpg
\(23\!\)

\(\begin{array}{lll} \text{p}_9^1 & = & \text{p}_{\text{p}_2^2}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1 \end{array}\)

\(\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!\) Riff 23 Big.jpg Rote 23 Big.jpg
\(25\!\)

\(\begin{array}{lll} \text{p}_3^2 & = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1} \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1} \end{array}\)

\(\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!\) Riff 25 Big.jpg Rote 25 Big.jpg
\(27\!\)

\(\begin{array}{lll} \text{p}_2^3 & = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1} \\[12pt] & = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1} \end{array}\)

\(\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!\) Riff 27 Big.jpg Rote 27 Big.jpg
\(32\!\)

\(\begin{array}{lll} \text{p}_1^5 & = & \text{p}_1^{\text{p}_3^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!\) Riff 32 Big.jpg Rote 32 Big.jpg
\(49\!\)

\(\begin{array}{lll} \text{p}_4^2 & = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1} \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1} \end{array}\)

\(\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!\) Riff 49 Big.jpg Rote 49 Big.jpg
\(53\!\)

\(\begin{array}{lll} \text{p}_{16}^1 & = & \text{p}_{\text{p}_1^4}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1 \end{array}\)

\(\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!\) Riff 53 Big.jpg Rote 53 Big.jpg
\(64\!\)

\(\begin{array}{lll} \text{p}_1^6 & = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1} \\[12pt] & = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1} \end{array}\)

\(\text{p}^{\text{p} \text{p}_{\text{p}}}\!\) Riff 64 Big.jpg Rote 64 Big.jpg
\(81\!\)

\(\begin{array}{lll} \text{p}_2^4 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2} \\[12pt] & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}} \end{array}\)

\(\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!\) Riff 81 Big.jpg Rote 81 Big.jpg
\(128\!\)

\(\begin{array}{lll} \text{p}_1^7 & = & \text{p}_1^{\text{p}_4^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!\) Riff 128 Big.jpg Rote 128 Big.jpg
\(256\!\)

\(\begin{array}{lll} \text{p}_1^8 & = & \text{p}_1^{\text{p}_1^3} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}} \end{array}\)

\(\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!\) Riff 256 Big.jpg Rote 256 Big.jpg
\(512\!\)

\(\begin{array}{lll} \text{p}_1^9 & = & \text{p}_1^{\text{p}_2^2} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}} \end{array}\)

\(\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!\) Riff 512 Big.jpg Rote 512 Big.jpg
\(65536\!\)

\(\begin{array}{lll} \text{p}_1^{16} & = & \text{p}_1^{\text{p}_1^4} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}} \end{array}\)

\(\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!\) Riff 65536 Big.jpg Rote 65536 Big.jpg

A062537

  • Nodes in riff (rooted index-functional forest) for n.
\(a(n) = \text{Number of Nodes in the Riff of}~ n\)

 


\(1\!\)


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

Riff 2 Big.jpg


\(\text{p}\!\)


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

Riff 3 Big.jpg


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


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

Riff 4 Big.jpg


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


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

Riff 5 Big.jpg


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


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

Riff 6 Big.jpg


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


\(a(6) ~=~ 3\)

Riff 7 Big.jpg


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


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

Riff 8 Big.jpg


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


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

Riff 9 Big.jpg


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


\(a(9) ~=~ 3\)

Riff 10 Big.jpg


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


\(a(10) ~=~ 4\)

Riff 11 Big.jpg


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


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

Riff 12 Big.jpg


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


\(a(12) ~=~ 4\)

Riff 13 Big.jpg


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


\(a(13) ~=~ 4\)

Riff 14 Big.jpg


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


\(a(14) ~=~ 4\)

Riff 15 Big.jpg


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


\(a(15) ~=~ 5\)

Riff 16 Big.jpg


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


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

Riff 17 Big.jpg


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


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

Riff 18 Big.jpg


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


\(a(18) ~=~ 4\)

Riff 19 Big.jpg


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


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

Riff 20 Big.jpg


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


\(a(20) ~=~ 5\)

Riff 21 Big.jpg


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


\(a(21) ~=~ 5\)

Riff 22 Big.jpg


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


\(a(22) ~=~ 5\)

Riff 23 Big.jpg


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


\(a(23) ~=~ 4\)

Riff 24 Big.jpg


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


\(a(24) ~=~ 5\)

Riff 25 Big.jpg


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


\(a(25) ~=~ 4\)

Riff 26 Big.jpg


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


\(a(26) ~=~ 5\)

Riff 27 Big.jpg


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


\(a(27) ~=~ 4\)

Riff 28 Big.jpg


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


\(a(28) ~=~ 5\)

Riff 29 Big.jpg


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


\(a(29) ~=~ 5\)

Riff 30 Big.jpg


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


\(a(30) ~=~ 6\)

Riff 31 Big.jpg


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


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

Riff 32 Big.jpg


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


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

Riff 33 Big.jpg


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


\(a(33) ~=~ 6\)

Riff 34 Big.jpg


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


\(a(34) ~=~ 5\)

Riff 35 Big.jpg


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


\(a(35) ~=~ 6\)

Riff 36 Big.jpg


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


\(a(36) ~=~ 5\)

Riff 37 Big.jpg


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


\(a(37) ~=~ 5\)

Riff 38 Big.jpg


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


\(a(38) ~=~ 5\)

Riff 39 Big.jpg


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


\(a(39) ~=~ 6\)

Riff 40 Big.jpg


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


\(a(40) ~=~ 6\)

Riff 41 Big.jpg


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


\(a(41) ~=~ 5\)

Riff 42 Big.jpg


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


\(a(42) ~=~ 6\)

Riff 43 Big.jpg


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


\(a(43) ~=~ 5\)

Riff 44 Big.jpg


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


\(a(44) ~=~ 6\)

Riff 45 Big.jpg


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


\(a(45) ~=~ 6\)

Riff 46 Big.jpg


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


\(a(46) ~=~ 5\)

Riff 47 Big.jpg


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


\(a(47) ~=~ 6\)

Riff 48 Big.jpg


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


\(a(48) ~=~ 5\)

Riff 49 Big.jpg


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


\(a(49) ~=~ 4\)

Riff 50 Big.jpg


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


\(a(50) ~=~ 5\)

Riff 51 Big.jpg


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


\(a(51) ~=~ 6\)

Riff 52 Big.jpg


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


\(a(52) ~=~ 6\)

Riff 53 Big.jpg


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


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

Riff 54 Big.jpg


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


\(a(54) ~=~ 5\)

Riff 55 Big.jpg


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


\(a(55) ~=~ 7\)

Riff 56 Big.jpg


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


\(a(56) ~=~ 6\)

Riff 57 Big.jpg


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


\(a(57) ~=~ 6\)

Riff 58 Big.jpg


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


\(a(58) ~=~ 6\)

Riff 59 Big.jpg


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


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

Riff 60 Big.jpg


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


\(a(60) ~=~ 7\)

A062860

  • Smallest j with n nodes in its riff (rooted index-functional forest).
\(a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}\)

 


\(1\!\)


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

Riff 2 Big.jpg


\(\text{p}\!\)


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

Riff 3 Big.jpg


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


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

Riff 5 Big.jpg


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


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

Riff 10 Big.jpg


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


\(a(4) ~=~ 10\)

Riff 15 Big.jpg


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


\(a(5) ~=~ 15\)

Riff 30 Big.jpg


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


\(a(6) ~=~ 30\)

Riff 55 Big.jpg


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


\(a(7) ~=~ 55\)

Riff 105 Big.jpg


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


\(a(8) ~=~ 105\)

Riff 165 Big.jpg


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


\(a(9) ~=~ 165\)

A109301

  • a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.
\(a(n) = \text{Rote Height of}~ n\)

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


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


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

Retrieved from "https://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Papers/Riffs_and_Rotes&oldid=106408"