Riffs in Numerical Order

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

 


\(1\!\)


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

 


\(\text{p}\!\)


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

Rotes in Numerical Order

 


\(1\!\)


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

 


\(\text{p}\!\)


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

Selected Sequences

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.
\(\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!\)
\(\text{Integer}\!\) \(\text{Factorization}\!\) \(\text{Notation}\!\) \(\text{Riff Digraph}\!\) \(\text{Rote Graph}\!\) \(\text{Traversal}\!\)
\(1\!\) \(1\!\)        
\(2\!\) \(\text{p}_1^1\!\) \(\text{p}\!\)     \(((~))\)
\(3\!\)

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

\(\text{p}_\text{p}\!\)     \((((~))(~))\)
\(4\!\)

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

\(\text{p}^\text{p}\!\)     \(((((~))))\)
\(5\!\)

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

\(\text{p}_{\text{p}_{\text{p}}}\!\)     \(((((~))(~))(~))\)
\(6\!\)

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

\(\text{p} \text{p}_{\text{p}}\!\)     \(((~))(((~))(~))\)
\(7\!\)

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

\(\text{p}_{\text{p}^{\text{p}}}\!\)     \((((((~))))(~))\)
\(8\!\)

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

\(\text{p}^{\text{p}_{\text{p}}}\!\)     \((((((~))(~))))\)
\(9\!\)

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

\(\text{p}_\text{p}^\text{p}\!\)     \((((~))(((~))))\)
\(16\!\)

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

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

A062504

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

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

\(\text{Prime Factorizations, Riffs, and Rotes}\!\)
\(\text{Integer}\!\) \(\text{Factorization}\!\) \(\text{Notation}\!\) \(\text{Riff Digraph}\!\) \(\text{Rote Graph}\!\)
\(1\!\) \(1\!\)      
\(2\!\) \(\text{p}_1^1\!\) \(\text{p}\!\)    
\(3\!\)

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

\(\text{p}_\text{p}\!\)    
\(4\!\)

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

\(\text{p}^\text{p}\!\)    
\(5\!\)

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

\(\text{p}_{\text{p}_{\text{p}}}\!\)    
\(6\!\)

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

\(\text{p} \text{p}_{\text{p}}\!\)    
\(7\!\)

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

\(\text{p}_{\text{p}^{\text{p}}}\!\)    
\(8\!\)

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

\(\text{p}^{\text{p}_{\text{p}}}\!\)    
\(9\!\)

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

\(\text{p}_\text{p}^\text{p}\!\)    
\(16\!\)

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

\(\text{p}^{\text{p}^{\text{p}}}\!\)    
\(10\!\)

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

\(\text{p} \text{p}_{\text{p}_{\text{p}}}\!\)    
\(11\!\)

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

\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)    
\(12\!\)

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

\(\text{p}^{\text{p}} \text{p}_{\text{p}}\!\)    
\(13\!\)

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

\(\text{p}_{\text{p} \text{p}_{\text{p}}}\!\)    
\(14\!\)

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

\(\text{p} \text{p}_{\text{p}^{\text{p}}}\!\)    
\(17\!\)

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

\(\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!\)    
\(18\!\)

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

\(\text{p} \text{p}_{\text{p}}^{\text{p}}\!\)    
\(19\!\)

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

\(\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!\)    
\(23\!\)

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

\(\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!\)    
\(25\!\)

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

\(\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!\)    
\(27\!\)

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

\(\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!\)    
\(32\!\)

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

\(\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!\)    
\(49\!\)

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

\(\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!\)    
\(53\!\)

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

\(\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!\)    
\(64\!\)

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

\(\text{p}^{\text{p} \text{p}_{\text{p}}}\!\)    
\(81\!\)

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

\(\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!\)    
\(128\!\)

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

\(\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!\)    
\(256\!\)

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

\(\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!\)    
\(512\!\)

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

\(\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!\)    
\(65536\!\)

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

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

A062537

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

 


\(1\!\)


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

 


\(\text{p}\!\)


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

A062860

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

 


\(1\!\)


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

 


\(\text{p}\!\)


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

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.
\(a(n) = \text{Rote Height of}~ n\)

 


\(1\!\)


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

 


\(\text{p}\!\)


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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

 


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


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