Directory:Jon Awbrey/Papers/Riffs and Rotes

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

$\text{Prime Factorizations, Riffs, and Rotes}\!$

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

$1\!$ $1\!$

$2\!$ $\text{p}_1^1\!$ $\text{p}\!$

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

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

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

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.

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

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