Riffs and Rotes

Riffs in Numerical Order

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

\(\begin{array}{l} \varnothing \\ 1 \end{array}\)	\(\text{p}\!\) \(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)	\(\text{p}_\text{p}\!\) \(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)	\(\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)	\(\text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)
\(\text{p} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)	\(\text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)	\(\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)	\(\text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)	\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)	\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)	\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)	\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)
\(\text{p}^{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)	\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)	\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)	\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)	\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)	\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)	\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)	\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)	\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)	\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)	\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)	\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)	\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)	\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)	\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)	\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)	\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)	\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)	\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)	\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)	\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)	\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)	\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)	\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Rotes in Numerical Order

\(\begin{array}{l} \varnothing \\ 1 \end{array}\)	\(\text{p}\!\) \(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)	\(\text{p}_\text{p}\!\) \(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)	\(\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)	\(\text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)
\(\text{p} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)	\(\text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)	\(\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)	\(\text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)	\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)	\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)	\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)	\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)
\(\text{p}^{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)	\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)	\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)	\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)	\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)	\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)	\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)	\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)	\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)	\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)	\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)	\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)	\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)	\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)	\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)	\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)	\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)	\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)	\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)	\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)	\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)	\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)	\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)	\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)	\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)	\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Selected Sequences

\(\text{Prime Factorizations, Riffs, and Rotes}\!\)

\(\text{Integer}\!\) \(\text{Factorization}\!\) \(\text{Notation}\!\) \(\text{Riff Digraph}\!\) \(\text{Rote Graph}\!\)

\(1\!\) \(1\!\)

\(2\!\) \(\text{p}_1^1\!\) \(\text{p}\!\)

\(3\!\) \(\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}\) \(\text{p}_\text{p}\!\) \(4\!\) \(\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}\) \(\text{p}^\text{p}\!\)

\(5\!\) \(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\) \(\text{p}_{\text{p}_{\text{p}}}\!\) \(6\!\) \(\begin{array}{lll} \text{p}_1^1 \text{p}_2^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 \end{array}\) \(\text{p} \text{p}_{\text{p}}\!\) \(7\!\) \(\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\) \(\text{p}_{\text{p}^{\text{p}}}\!\) \(8\!\) \(\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}\) \(\text{p}^{\text{p}_{\text{p}}}\!\) \(9\!\) \(\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\) \(\text{p}_\text{p}^\text{p}\!\) \(16\!\) \(\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}\) \(\text{p}^{\text{p}^{\text{p}}}\!\)

\(10\!\) \(\begin{array}{lll} \text{p}_1^1 \text{p}_3^1 & = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\) \(\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(11\!\) \(\begin{array}{lll} \text{p}_5^1 & = & \text{p}_{\text{p}_3^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 \end{array}\) \(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(12\!\) \(\begin{array}{lll} \text{p}_1^2 \text{p}_2^1 & = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1 \end{array}\) \(\text{p}^{\text{p}} \text{p}_{\text{p}}\!\) \(13\!\) \(\begin{array}{lll} \text{p}_6^1 & = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1 \\[12pt] & = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1 \end{array}\) \(\text{p}_{\text{p} \text{p}_{\text{p}}}\!\) \(14\!\) \(\begin{array}{lll} \text{p}_1^1 \text{p}_4^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1 \\[12pt] & = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\) \(\text{p} \text{p}_{\text{p}^{\text{p}}}\!\) \(17\!\) \(\begin{array}{lll} \text{p}_7^1 & = & \text{p}_{\text{p}_4^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \end{array}\) \(\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!\) \(18\!\) \(\begin{array}{lll} \text{p}_1^1 \text{p}_2^2 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\) \(\text{p} \text{p}_{\text{p}}^{\text{p}}\!\) \(19\!\) \(\begin{array}{lll} \text{p}_8^1 & = & \text{p}_{\text{p}_1^3}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1 \end{array}\) \(\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!\) \(23\!\) \(\begin{array}{lll} \text{p}_9^1 & = & \text{p}_{\text{p}_2^2}^1 \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1 \end{array}\) \(\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!\) \(25\!\) \(\begin{array}{lll} \text{p}_3^2 & = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1} \\[12pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1} \end{array}\) \(\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!\) \(27\!\) \(\begin{array}{lll} \text{p}_2^3 & = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1} \\[12pt] & = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1} \end{array}\) \(\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!\) \(32\!\) \(\begin{array}{lll} \text{p}_1^5 & = & \text{p}_1^{\text{p}_3^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1} \end{array}\) \(\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!\) \(49\!\) \(\begin{array}{lll} \text{p}_4^2 & = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1} \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1} \end{array}\) \(\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!\) \(53\!\) \(\begin{array}{lll} \text{p}_{16}^1 & = & \text{p}_{\text{p}_1^4}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1 \\[12pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1 \end{array}\) \(\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!\) \(64\!\) \(\begin{array}{lll} \text{p}_1^6 & = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1} \\[12pt] & = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1} \end{array}\) \(\text{p}^{\text{p} \text{p}_{\text{p}}}\!\) \(81\!\) \(\begin{array}{lll} \text{p}_2^4 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2} \\[12pt] & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}} \end{array}\) \(\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!\) \(128\!\) \(\begin{array}{lll} \text{p}_1^7 & = & \text{p}_1^{\text{p}_4^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1} \end{array}\) \(\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!\) \(256\!\) \(\begin{array}{lll} \text{p}_1^8 & = & \text{p}_1^{\text{p}_1^3} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}} \end{array}\) \(\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!\) \(512\!\) \(\begin{array}{lll} \text{p}_1^9 & = & \text{p}_1^{\text{p}_2^2} \\[12pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}} \end{array}\) \(\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!\) \(65536\!\) \(\begin{array}{lll} \text{p}_1^{16} & = & \text{p}_1^{\text{p}_1^4} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}} \\[12pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}} \end{array}\) \(\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!\)

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.

• OEIS Wiki Entry for A061396.

A062504

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

• OEIS Wiki Entry for A062504.

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

A062537

• Nodes in riff (rooted index-functional forest) for n.

• OEIS Wiki Entry for A062537.

\(a(n) = \text{Number of Nodes in the Riff of}~ n\)

\(a(1) ~=~ 0\)	\(\text{p}\!\) \(a(2) ~=~ 1\)	\(\text{p}_\text{p}\!\) \(a(3) ~=~ 2\)	\(\text{p}^\text{p}\!\) \(a(4) ~=~ 2\)	\(\text{p}_{\text{p}_{\text{p}}}\!\) \(a(5) ~=~ 3\)
\(\text{p} \text{p}_{\text{p}}\!\) \(a(6) ~=~ 3\)	\(\text{p}_{\text{p}^{\text{p}}}\!\) \(a(7) ~=~ 3\)	\(\text{p}^{\text{p}_{\text{p}}}\!\) \(a(8) ~=~ 3\)	\(\text{p}_\text{p}^\text{p}\!\) \(a(9) ~=~ 3\)	\(\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(10) ~=~ 4\)
\(\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!\) \(a(11) ~=~ 4\)	\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(a(12) ~=~ 4\)	\(\text{p}_{\text{p} \text{p}_{\text{p}}}\!\) \(a(13) ~=~ 4\)	\(\text{p} \text{p}_{\text{p}^{\text{p}}}\!\) \(a(14) ~=~ 4\)	\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(15) ~=~ 5\)
\(\text{p}^{\text{p}^{\text{p}}}\!\) \(a(16) ~=~ 3\)	\(\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!\) \(a(17) ~=~ 4\)	\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(18) ~=~ 4\)	\(\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!\) \(a(19) ~=~ 4\)	\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(20) ~=~ 5\)
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(21) ~=~ 5\)	\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(22) ~=~ 5\)	\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(23) ~=~ 4\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(a(24) ~=~ 5\)	\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(25) ~=~ 4\)
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(26) ~=~ 5\)	\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(27) ~=~ 4\)	\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(28) ~=~ 5\)	\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(29) ~=~ 5\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(30) ~=~ 6\)
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(a(31) ~=~ 5\)	\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(a(32) ~=~ 4\)	\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(33) ~=~ 6\)	\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(34) ~=~ 5\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(35) ~=~ 6\)
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(36) ~=~ 5\)	\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(a(37) ~=~ 5\)	\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(38) ~=~ 5\)	\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(39) ~=~ 6\)	\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(a(40) ~=~ 6\)
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(a(41) ~=~ 5\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(42) ~=~ 6\)	\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(a(43) ~=~ 5\)	\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(44) ~=~ 6\)	\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(45) ~=~ 6\)
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(46) ~=~ 5\)	\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(47) ~=~ 6\)	\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(a(48) ~=~ 5\)	\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(a(49) ~=~ 4\)	\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(50) ~=~ 5\)
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(51) ~=~ 6\)	\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(52) ~=~ 6\)	\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(a(53) ~=~ 4\)	\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(54) ~=~ 5\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(55) ~=~ 7\)
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(56) ~=~ 6\)	\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(57) ~=~ 6\)	\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(58) ~=~ 6\)	\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(a(59) ~=~ 5\)	\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(60) ~=~ 7\)

A062860

• Smallest j with n nodes in its riff (rooted index-functional forest).

• OEIS Wiki Entry for A062860.

\(a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}\)

\(1\!\) \(a(0) ~=~ 1\)	\(\text{p}\!\) \(a(1) ~=~ 2\)	\(\text{p}_\text{p}\!\) \(a(2) ~=~ 3\)	\(\text{p}_{\text{p}_{\text{p}}}\!\) \(a(3) ~=~ 5\)	\(\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(4) ~=~ 10\)
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(5) ~=~ 15\)	\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(6) ~=~ 30\)	\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(7) ~=~ 55\)	\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(8) ~=~ 105\)	\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(9) ~=~ 165\)