User:Jon Awbrey/WORKSPACE

From OeisWiki

I thought I might start by upgrading the graphics for some of my sequences from ASCII to EPS, GIF, JPG, PNG, whatever works best here — maybe SVG if I can find an easy enough graphics package that works for that. Jon Awbrey 02:30, 31 October 2009 (UTC)

Other Workspaces: User:Jon Awbrey/ANIMATION; User:Jon Awbrey/GRAPHICS; User:Jon Awbrey/SVG; User:Jon Awbrey/TABLES; User:Jon Awbrey/VIDEOS

1 A061396
- 1.1 Wiki + TeX + JPEG
- 1.2 ASCII
2 A062504
- 2.1 TeX Array
- 2.2 JPEG
- 2.3 ASCII
3 A062537
- 3.1 Wiki + TeX + JPEG
4 A062860
- 4.1 Wiki + TeX + JPEG
5 A106177
- 5.1 Primal Codes of Finite Partial Functions on Positive Integers
- 5.2 Wiki Table
- 5.3 Wiki + TeX
- 5.4 ASCII
6 A106178
- 6.1 Wiki Table
- 6.2 TeX Smallmatrix
- 6.3 ASCII
7 A108352
- 7.1 Links
- 7.2 TeX Array
- 7.3 ASCII
8 A108353
- 8.1 TeX Array
- 8.2 ASCII
9 A108370
10 A108371
- 10.1 Wiki Table
- 10.2 ASCII
11 A108372
12 A108373
13 A108374
- 13.1 TeX Array
- 13.2 ASCII
14 A109297
- 14.1 TeX Array
- 14.2 ASCII
15 A109298
- 15.1 TeX Array
- 15.2 ASCII
16 A109299
- 16.1 TeX Array
- 16.2 ASCII
17 A109300
- 17.1 JPEG
- 17.2 ASCII
18 A109301
- 18.1 Example
- 18.2 JPEG
- 18.3 ASCII
19 A111788
20 A111789
21 A111790
22 A111791
- 22.1 TeX Array
- 22.2 Wiki Table
- 22.3 ASCII
23 A111792
- 23.1 TeX Array
- 23.2 ASCII
24 A111793
- 24.1 TeX Array
- 24.2 ASCII
25 A111794
- 25.1 TeX Array
- 25.2 Wiki Table
- 25.3 ASCII
26 A111795
- 26.1 JPEG
- 26.2 ASCII
27 A111796
- 27.1 TeX Array
- 27.2 ASCIII
28 A111797
- 28.1 TeX Array
- 28.2 ASCII
29 A111798
- 29.1 TeX Array
- 29.2 ASCII
30 A111799
- 30.1 TeX Array
- 30.2 ASCII
31 A111800
- 31.1 TeX + JPEG
- 31.2 ASCII
32 A111801
33 A112095
- 33.1 TeX Array
- 33.2 ASCII
34 A112096
- 34.1 TeX Array
- 34.2 ASCII
35 A112480
- 35.1 TeX Array
- 35.2 ASCII
36 A112481
- 36.1 TeX Array
- 36.2 ASCII
37 A112846
38 A112868
- 38.1 TeX Array
- 38.2 ASCII
39 A112869
- 39.1 TeX Array
- 39.2 ASCII
40 A112870
- 40.1 TeX Array
- 40.2 ASCII
41 A112871
- 41.1 TeX Array
- 41.2 ASCII
42 A112872
43 A113197
- 43.1 TeX Array
- 43.2 ASCII
44 A113198
- 44.1 TeX Array
- 44.2 ASCII
45 A113199
- 45.1 TeX Array
- 45.2 ASCII
46 A113200
- 46.1 TeX Array
- 46.2 ASCII

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.

Wiki + TeX + JPEG

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

$\text{Integer}\!$

$\text{Factorization}\!$

$\text{Notation}\!$

$\text{Riff Digraph}\!$

$\text{Rote Graph}\!$

$\text{Traversal}\!$

$1\!$

$2\!$

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

$\text{p}\!$

$((~))$

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

$5\!$	$\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[10pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}$	$\text{p}_{\text{p}_{\text{p}}}\!$	$((((~))(~))(~))$
$6\!$	$\begin{array}{lll} \text{p}_1^1 \text{p}_2^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 \end{array}$	$\text{p} \text{p}_{\text{p}}\!$	$((~))(((~))(~))$
$7\!$	$\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[10pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}$	$\text{p}_{\text{p}^{\text{p}}}\!$	$(((((~))))(~))$
$8\!$	$\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[10pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}$	$\text{p}^{\text{p}_{\text{p}}}\!$	$(((((~))(~))))$
$9\!$	$\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}$	$\text{p}_\text{p}^\text{p}\!$	$(((~))(((~))))$
$16\!$	$\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[10pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}$	$\text{p}^{\text{p}^{\text{p}}}\!$	$((((((~))))))$

ASCII

Illustration of initial terms of A061396
Jon Awbrey (jawbrey(AT)oakland.edu)

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_(p_p)   @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p_(p^p)   @     o      @            ((((())))())
|                                        ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =                    /   \       |
|           p_1^p_(p_1)       p^p_p     @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =                  /           |
|           p_(p_1)^(p_1)     p_p^p     @            @            ((())((())))
|                                        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =                       ^          o---o
|           p_1^(p_1^2) =                /           |
|           p_1^(p_1^p_1)     p^(p^p)   @            @            (((((())))))
|
o--------------------------------------------------------------------------------

Further Comments:

Here are a couple more pages from my notes,
where it looks like I first arrived at the
generating function, and also carried out
some brute force enumerations of riffs.

I am going to experiment with a different way of
transcribing indices and powers into a plaintext.

|                jj
|              p<
|      j      /  ji
|    p<     p<         etc.
|      i      \  ij
|              p<
|                ii

-------------------------------------------------------

1978-11-06

Generating Function

| R(x) = 1 + x + 2x^2 + ...
|
|      =   1 + x.x^0 (1 + x + 2x^2 + ...)
|        . 1 + x.x^1 (1 + x + 2x^2 + ...)
|        . 1 + x.x^2 (1 + x + 2x^2 + ...)
|        . 1 + x.x^2 (1 + x + 2x^2 + ...)
|        . ...
|
|      = 1 + x + 2x^2 + ...
|
| Product over (i = 0 to infinity) of (1 + x.x^i.R(x))^R_i  =  R(x)

-------------------------------------------------------

1978-11-10

Brute force enumeration of R_n

| 4 p's
|
|       p
|     p<        p_p                 p                    p
|   p<        p<        p p_p     p<_p     p_p_p     p_p<
| p<        p<        p<        p<       p<        p<
|
|
|       p
|     p<        p_p                 p                    p
| p_p<      p_p<      p<        p_p<_p   p_p_p_p   p_p_p<
|                       p p_p
|
|
|     p
|   p<        p_p       p         p        p           p
| p<        p<        p<        p<       p<  p<    p p<
|   p         p         p_p       p^p          p       p
|
|
| p p_p_p   p p<
|               p^p
|

Altogether, 20 riffs of weight 4.

| o---------------------o---------------------o---------------------o
| | 3                   | 4                   | 5                   |
| o---------------------o---------------------o---------------------|
| | // // 2             | 10, 3, 1, 6         | 36, 10, 2, 3, 2, 20 |
| o---------------------o---------------------o---------------------|
| |                     | 0^1 4^1,            |                     |
| |                     | 1^1 3^1,            |                     |
| |                     | 2^2,                |                     |
| |                     | 4^1 0^1             |                     |
| o---------------------o---------------------o---------------------o
| | 6                   | 20                  | 73                  |
| o---------------------o---------------------o---------------------o
|

-------------------------------------------------------

Here are the number values of the riffs on 4 nodes:

o----------------------------------------------------------------------
|
|       p
|     p<        p_p                 p                    p
|   p<        p<        p p_p     p<_p     p_p_p     p_p<
| p<        p<        p<        p<       p<        p<
|
| 2^16      2^8       2^6       2^9      2^5       2^7
| 65536     256       64        512      32        128
o----------------------------------------------------------------------
|
|       p
|     p<        p_p                 p                    p
| p_p<      p_p<      p<        p_p<_p   p_p_p_p   p_p_p<
|                       p p_p
|
| p_16      p_8       p_6       p_9      p_5       p_7
| 53        19        13        23       11        17
o----------------------------------------------------------------------
|
|     p
|   p<        p_p       p         p                    p
| p<        p<        p<        p<       p^p p_p   p p<
|   p         p         p_p       p^p                  p
|
| 3^4       3^3       5^2       7^2
| 81        27        25        49       12        18
o----------------------------------------------------------------------
|
| p p_p_p   p p<
|               p^p
|
| 10        14 
o----------------------------------------------------------------------

For ease of reference, I include the previous table
of smaller riffs and rotes, redone in the new style.

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_p_p     @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p<        @     o      @            ((((())))())
|                               p^p      ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =           p_p      /   \       |
|           p_1^p_(p_1)       p<        @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =         p        /           |
|           p_(p_1)^(p_1)     p<        @            @            ((())((())))
|                               p        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =               p       ^          o---o
|           p_1^(p_1^2) =       p<       /           |
|           p_1^(p_1^p_1)     p<        @            @            (((((())))))
|
o--------------------------------------------------------------------------------

(later)

Expanded version of first table:

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_p_p     @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p<        @     o      @            ((((())))())
|                               p^p      ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =           p_p      /   \       |
|           p_1^p_(p_1)       p<        @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =         p        /           |
|           p_(p_1)^(p_1)     p<        @            @            ((())((())))
|                               p        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =               p       ^          o---o
|           p_1^(p_1^2) =       p<       /           |
|           p_1^(p_1^p_1)     p<        @            @            (((((())))))
|
o--------------------------------------------------------------------------------

o================================================================================
|
|       p
|     p<        p          p_p         p
|   p<        p<_p       p<        p_p<      p p_p     p_p_p
| p<        p<         p<        p<        p<        p<
|
| 2^16      2^9        2^8       2^7       2^6       2^5
| 65536     512        256       128       64        32
|
o--------------------------------------------------------------------------------
|
|       p
|     p<        p          p_p         p
| p_p<      p_p<_p     p_p<      p_p_p<    p<        p_p_p_p
|                                            p p_p
|
| p_16      p_9        p_8       p_7       p_6       p_5
| 53        23         19        17        13        11
|
o--------------------------------------------------------------------------------
|
|   p^p       p_p        p         p
| p<        p<         p<        p<
|   p         p          p^p       p_p
|
| 3^4       3^3        7^2       5^2
| 81        27         49        25
|
o--------------------------------------------------------------------------------
|
|     p
| p p<      p p<       p^p p_p   p p_p_p
|     p         p^p
|
| 18        14         12        10
|
o================================================================================

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

k | natural numbers n such that |riff(n)| = k
--o------------------------------------------------
0 | 1;
1 | 2;
2 | 3, 4;
3 | 5, 6, 7, 8, 9, 16;
4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27,
  | 32, 49, 53, 64, 81, 128, 256, 512, 65536;

The natural number values for the riffs with
at most 3 pts are as follows (@'s are roots):

|                  o       o  o       o
|                  |       ^  |       ^
|                  v       |  v       |
|            o  o  o    o  o  o  o o  o
|            |  ^  |    |  |  ^  | ^  ^
|            v  |  v    v  v  |  v/   |
| Riff:   @; @, @; @, @ @, @, @, @,   @;
|
| Value:  2; 3, 4; 5,  6 , 7, 8, 9,  16;

---------------------------------------------------

1, 2, 3, 4, 5, 6, 7, 8, 9, 16,
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536,

---------------------------------------------------

1; 2; 3, 4; 5, 6, 7, 8, 9, 16;
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536;

---------------------------------------------------

A062504

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

TeX Array

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

JPEG

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

$\text{Integer}\!$

$\text{Factorization}\!$

$\text{Notation}\!$

$\text{Riff Digraph}\!$

$\text{Rote Graph}\!$

$1\!$

$2\!$

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

$\text{p}\!$

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

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

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

ASCII

 Example

    * k | natural numbers n such that |riff(n)| = k
    * 0 | 1;
    * 1 | 2;
    * 2 | 3, 4;
    * 3 | 5, 6, 7, 8, 9, 16;
    * 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536;
    * The natural number values for the riffs with at most 3 pts are as follows (x = root):
    * .................o.......o..o.......o
    * .................|.......^..|.......^
    * .................v.......|..v.......|
    * ...........o..o..o....o..o..o..o.o..o
    * ...........|..^..|....|..|..^..|.^..^
    * ...........v..|..v....v..v..|..v/...|
    * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
    * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;

A062537

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

Wiki + TeX + JPEG

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

A062860

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

Wiki + TeX + JPEG

$a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}$
$1\!$ $a(0) ~=~ 1$	$\text{p}\!$ $a(1) ~=~ 2$	$\text{p}_\text{p}\!$ $a(2) ~=~ 3$	$\text{p}_{\text{p}_{\text{p}}}\!$ $a(3) ~=~ 5$	$\text{p} \text{p}_{\text{p}_{\text{p}}}\!$ $a(4) ~=~ 10$
$\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!$ $a(5) ~=~ 15$	$\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$ $a(6) ~=~ 30$	$\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(7) ~=~ 55$	$\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$ $a(8) ~=~ 105$	$\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$ $a(9) ~=~ 165$

A106177

Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function.

Primal Codes of Finite Partial Functions on Positive Integers

$\begin{array}{rcl} 1 & = & \varnothing \\ 2 & = & 1\!:\!1 \\ 3 & = & 2\!:\!1 \\ 4 & = & 1\!:\!2 \\ 5 & = & 3\!:\!1 \\ 6 & = & 1\!:\!1 ~~ 2\!:\!1 \\ 7 & = & 4\!:\!1 \\ 8 & = & 1\!:\!3 \\ 9 & = & 2\!:\!2 \\ 10 & = & 1\!:\!1 ~~ 3\!:\!1 \\ 11 & = & 5\!:\!1 \\ 12 & = & 1\!:\!2 ~~ 2\!:\!1 \\ 13 & = & 6\!:\!1 \\ 14 & = & 1\!:\!1 ~~ 4\!:\!1 \\ 15 & = & 2\!:\!1 ~~ 3\!:\!1 \\ 16 & = & 1\!:\!4 \\ 17 & = & 7\!:\!1 \\ 18 & = & 1\!:\!1 ~~ 2\!:\!2 \\ 19 & = & 8\!:\!1 \\ 20 & = & 1\!:\!2 ~~ 3\!:\!1 \end{array}$

Wiki Table

									1		1
								2		1		2
							3		1		1		3
						4		1		2		1		4
					5		1		3		1		1		5
				6		1		1		1		4		1		6
			7		1		5		2		9		1		1		7
		8		1		6		1		1		1		2		1		8
	9		1		7		1		25		1		3		1		1		9
10		1		1		1		36		1		2		1		8		1		10

Wiki + TeX

Smallmatrix

$\begin{smallmatrix} & & & & & & & & & {\color{red}1} & & {\color{red}1} \\ & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} \\ & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} \\ & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} \\ & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} \\ & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} \\ & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} \\ & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} \\ & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} \\ {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} \end{smallmatrix}$

Array

$\begin{array}{*{21}{c}} & & & & & & & & & {\color{red}1} & & {\color{red}1} \\ & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} \\ & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} \\ & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} \\ & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} \\ & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} \\ & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} \\ & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} \\ & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} \\ {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} \end{array}$

Matrix

$\begin{matrix} n \circ m \\ 1 ~/~\backslash~ 1 \\ 2 ~/~ 1 ~\backslash~ 2 \\ 3 ~/~ 1 \cdot 1 ~\backslash~ 3 \\ 4 ~/~ 1 \cdot 2 \cdot 1 ~\backslash~ 4 \\ 5 ~/~ 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 5 \\ 6 ~/~ 1 \cdot 1 \cdot 1 \cdot 4 \cdot 1 ~\backslash~ 6 \\ 7 ~/~ 1 \cdot 5 \cdot 2 \cdot 9 \cdot 1 \cdot 1 ~\backslash~ 7 \\ 8 ~/~ 1 \cdot 6 \cdot 1 \cdot 1 \cdot 1 \cdot 2 \cdot 1 ~\backslash~ 8 \\ 9 ~/~ 1 \cdot 7 \cdot 1 \cdot 25\cdot 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 9 \\ 10 ~/~ 1 \cdot 1 \cdot 1 \cdot 36\cdot 1 \cdot 2 \cdot 1 \cdot 8 \cdot 1 ~\backslash~ 10 \end{matrix}$

ASCII

 Example

    *                      n o m
    *                       \ /
    *                      1 . 1
    *                     \ / \ /
    *                    2 . 1 . 2
    *                   \ / \ / \ /
    *                  3 . 1 . 1 . 3
    *                 \ / \ / \ / \ /
    *                4 . 1 . 2 . 1 . 4
    *               \ / \ / \ / \ / \ /
    *              5 . 1 . 3 . 1 . 1 . 5
    *             \ / \ / \ / \ / \ / \ /
    *            6 . 1 . 1 . 1 . 4 . 1 . 6
    *           \ / \ / \ / \ / \ / \ / \ /
    *          7 . 1 . 5 . 2 . 9 . 1 . 1 . 7
    *         \ / \ / \ / \ / \ / \ / \ / \ /
    *        8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8
    *       \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *      9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9
    *     \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *   10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10
    *
    * Primal codes of finite partial functions on positive integers:
    * 1 = { }
    * 2 = 1:1
    * 3 = 2:1
    * 4 = 1:2
    * 5 = 3:1
    * 6 = 1:1 2:1
    * 7 = 4:1
    * 8 = 1:3
    * 9 = 2:2
    * 10 = 1:1 3:1
    * 11 = 5:1
    * 12 = 1:2 2:1
    * 13 = 6:1
    * 14 = 1:1 4:1
    * 15 = 2:1 3:1
    * 16 = 1:4
    * 17 = 7:1
    * 18 = 1:1 2:2
    * 19 = 8:1
    * 20 = 1:2 3:1

A106178

Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function, but omitting the trivial values of 1 at the margins of the table.

Wiki Table

															1		1
														2		·		2
													3		·		·		3
												4		·		2		·		4
											5		·		3		1		·		5
										6		·		1		1		4		·		6
									7		·		5		2		9		1		·		7
								8		·		6		1		1		1		2		·		8
							9		·		7		1		25		1		3		1		·		9
						10		·		1		1		36		1		2		1		8		·		10
					11		·		1		1		49		1		5		1		27		1		·		11
				12		·		10		3		1		1		6		1		1		1		2		·		12
			13		·		11		1		1		2		7		1		125		4		3		1		·		13
		14		·		3		1		100		1		1		1		216		1		1		1		4		·		14
	15		·		13		2		121		1		3		1		343		1		5		1		9		1		·		15
16		·		14		1		9		1		10		1		1		1		6		1		2		1		2		·		16

TeX Smallmatrix

$\begin{smallmatrix} &&&&&&&&&&&&&&& {\color{red}1} && {\color{red}1} \\ &&&&&&&&&&&&&& {\color{red}2} && \cdot & & {\color{red}2} \\ &&&&&&&&&&&&& {\color{red}3} && \cdot && \cdot && {\color{red}3} \\ &&&&&&&&&&&& {\color{red}4} && \cdot && 2 && \cdot && {\color{red}4} \\ &&&&&&&&&&& {\color{red}5} && \cdot && 3 && 1 && \cdot && {\color{red}5} \\ &&&&&&&&&& {\color{red}6} && \cdot && 1 && 1 && 4 && \cdot && {\color{red}6} \\ &&&&&&&&& {\color{red}7} && \cdot && 5 && 2 && 9 && 1 && \cdot && {\color{red}7} \\ &&&&&&&& {\color{red}8} && \cdot && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}8} \\ &&&&&&& {\color{red}9} && \cdot && 7 && 1 && 25 && 1 && 3 && 1 && \cdot && {\color{red}9} \\ &&&&&& {\color{red}10} && \cdot && 1 && 1 && 36 && 1 && 2 && 1 && 8 && \cdot && {\color{red}10} \\ &&&&& {\color{red}11} && \cdot && 1 && 1 && 49 && 1 && 5 && 1 && 27 && 1 && \cdot && {\color{red}11} \\ &&&& {\color{red}12} && \cdot && 10 && 3 && 1 && 1 && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}12} \\ &&& {\color{red}13} && \cdot && 11 && 1 && 1 && 2 && 7 && 1 && 125 && 4 && 3 && 1 && \cdot && {\color{red}13} \\ && {\color{red}14} && \cdot && 3 && 1 && 100 && 1 && 1 && 1 && 216 && 1 && 1 && 1 && 4 && \cdot && {\color{red}14} \\ & {\color{red}15} && \cdot && 13 && 2 && 121 && 1 && 3 && 1 && 343 && 1 && 5 && 1 && 9 && 1 && \cdot && {\color{red}15} \\ {\color{red}16} && \cdot && 14 && 1 && 9 && 1 && 10 && 1 && 1 && 1 && 6 && 1 && 2 && 1 && 2 && \cdot && {\color{red}16} \end{smallmatrix}$

ASCII

 Example

    *                                   n o m
    *                                    \ /
    *                                   1 . 1
    *                                  \ / \ /
    *                                 2 .   . 2
    *                                \ / \ / \ /
    *                               3 .   .   . 3
    *                              \ / \ / \ / \ /
    *                             4 .   . 2 .   . 4
    *                            \ / \ / \ / \ / \ /
    *                           5 .   . 3 . 1 .   . 5
    *                          \ / \ / \ / \ / \ / \ /
    *                         6 .   . 1 . 1 . 4 .   . 6
    *                        \ / \ / \ / \ / \ / \ / \ /
    *                       7 .   . 5 . 2 . 9 . 1 .   . 7
    *                      \ / \ / \ / \ / \ / \ / \ / \ /
    *                     8 .   . 6 . 1 . 1 . 1 . 2 .   . 8
    *                    \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *                   9 .   . 7 . 1 . 25. 1 . 3 . 1 .   . 9
    *                  \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *                10 .   . 1 . 1 . 36. 1 . 2 . 1 . 8 .   . 10
    *                \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *              11 .   . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 .   . 11
    *              \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *            12 .   . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 .   . 12
    *            \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *          13 .   . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 .   . 13
    *          \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *        14 .   . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 .   . 14
    *        \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *      15 .   . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 .   . 15
    *      \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *    16 .   . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 .   . 16

A108352

a(n) = primal code characteristic of n, which is the least positive integer, if any, such that (n o)^k = 1, otherwise equal to 0. Here "o" denotes the primal composition operator, as illustrated in A106177 and A108371, and (n o)^k = n o … o n, with k occurrences of n.

Links

Jon Awbrey, Primal Code Characteristic, n = 1 to 1000
Jon Awbrey, Primal Code Characteristic, n = 1001 to 2000
Jon Awbrey, Primal Code Characteristic, n = 2001 to 3000

TeX Array

$\begin{array}{*{10}{l}} a(1) & = & 1 & \text{because} & (\circ~ 1)^1 & = & (\circ~ \varnothing)^1 & = & 1. \\ a(2) & = & 0 & \text{because} & (\circ~ 2)^k & = & (\circ~ 1\!:\!1)^k & = & 2, & \text{for all}~ k > 0. \\ a(3) & = & 2 & \text{because} & (\circ~ 3)^2 & = & (\circ~ 2\!:\!1)^2 & = & 1. \\ a(4) & = & 2 & \text{because} & (\circ~ 4 )^2 & = & (\circ~ 1\!:\!2)^2 & = &1. \\ a(5) & = & 2 & \text{because} & (\circ~ 5)^2 & = & (\circ~ 3\!:\!1)^2 & = & 1. \\ a(6) & = & 0 & \text{because} & (\circ~ 6)^k & = & (\circ~ 1\!:\!1 ~~ 2\!:\!1)^k & = & 6, & \text{for all}~ k > 0. \\ a(7) & = & 2 & \text{because} & (\circ~ 7)^2 & = & (\circ~ 4\!:\!1)^1 & = & 1. \\ a(8) & = & 2 & \text{because} & (\circ~ 8)^2 & = & (\circ~ 1\!:\!3)^1 & = & 1. \\ a(9) & = & 0 & \text{because} & (\circ~ 9)^k & = & (\circ~ 2\!:\!2)^k & = & 9, & \text{for all}~ k > 0. \\ a(10) & = & 0 & \text{because} & (\circ~ 10)^k & = & (\circ~ 1\!:\!1 ~~ 3\!:\!1)^k & = & 10, & \text{for all}~ k > 0. \end{array}$

ASCII

Example

    * a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1.
    * a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k.
    * a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1.
    * a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1.
    * a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1.
    * a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
    * a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1.
    * a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1.
    * a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k.
    * a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.
    * Detail of calculation for compositional powers of 12:
    * (12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
    * (12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12
    * Detail of calculation for compositional powers of 20:
    * (20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
    * (20 o)^3 = (3:2) o (1:2 3:1) = 1

A108353

For each nonnegative integer n, a(n) is the smallest positive integer j whose primal code characteristic is n, that is, the smallest j such that A108352(j) = n.

TeX Array

$\begin{array}{rcl||l} 2 & = & 1\!:\!1 & 1 \mapsto 1 ~\text{(infinite loop)} \\ 1 & = & \varnothing & 1 \\ 3 & = & 2\!:\!1 & 2 \mapsto 1 \\ 20 & = & 1\!:\!2 ~~ 3\!:\!1 & 3 \mapsto 1 \mapsto 2 \\ 756 & = & 1\!:\!2 ~~ 2\!:\!3 ~~ 4\!:\!1 & 4 \mapsto 1 \mapsto 2 \mapsto 3 \\ 178200 & = & 1\!:\!3 ~~ 2\!:\!4 ~~ 3\!:\!2 ~~ 5\!:\!1 & 5 \mapsto 1 \mapsto 3 \mapsto 2 \mapsto 4 \end{array}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have the following table:
    * Primal Functions and Functional Digraphs for a(0) to a(5)
    *       2 = 1:1             || 1 -> 1 (infinite loop)
    *       1 = { }             || 1
    *       3 = 2:1             || 2 -> 1
    *      20 = 1:2 3:1         || 3 -> 1 -> 2
    *     756 = 1:2 2:3 4:1     || 4 -> 1 -> 2 -> 3
    *  178200 = 1:3 2:4 3:2 5:1 || 5 -> 1 -> 3 -> 2 -> 4

A108370

Numbers whose primal code characteristic = 0, that is, positive n for which A108352(n) = 0.

A108371

Table of primal compositional powers (n o)^k, where "o" denotes the primal composition operator, as illustrated in sequence A106177, and where (n o)^k = n o … o n, with k occurrences of n.

Wiki Table

															1		1
														2		1		2
													3		2		1		3
												4		3		2		1		4
											5		4		1		2		1		5
										6		5		1		1		2		1		6
									7		6		1		1		1		2		1		7
								8		7		6		1		1		1		2		1		8
							9		8		1		6		1		1		1		2		1		9
						10		9		1		1		6		1		1		1		2		1		10
					11		10		9		1		1		6		1		1		1		2		1		11
				12		11		10		9		1		1		6		1		1		1		2		1		12
			13		12		1		10		9		1		1		6		1		1		1		2		1		13
		14		13		18		1		10		9		1		1		6		1		1		1		2		1		14
	15		14		1		12		1		10		9		1		1		6		1		1		1		2		1		15
16		15		14		1		18		1		10		9		1		1		6		1		1		1		2		1		16

ASCII

 Example

    * Table: T(n,k) = (n o)^k
    *                                  T(n,k)
    *                                    \ /
    *                                   1 . 1
    *                                  \ / \ /
    *                                 2 . 1 . 2
    *                                \ / \ / \ /
    *                               3 . 2 . 1 . 3
    *                              \ / \ / \ / \ /
    *                             4 . 3 . 2 . 1 . 4
    *                            \ / \ / \ / \ / \ /
    *                           5 . 4 . 1 . 2 . 1 . 5
    *                          \ / \ / \ / \ / \ / \ /
    *                         6 . 5 . 1 . 1 . 2 . 1 . 6
    *                        \ / \ / \ / \ / \ / \ / \ /
    *                       7 . 6 . 1 . 1 . 1 . 2 . 1 . 7
    *                      \ / \ / \ / \ / \ / \ / \ / \ /
    *                     8 . 7 . 6 . 1 . 1 . 1 . 2 . 1 . 8
    *                    \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *                   9 . 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 9
    *                  \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *                10 . 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 10
    *                \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *              11 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 11
    *              \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *            12 . 11. 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 12
    *            \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *          13 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 13
    *          \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *        14 . 13. 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 14
    *        \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *      15 . 14. 1 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 15
    *      \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *    16 . 15. 14. 1 . 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 16

A108372

Numbers whose primal code characteristic = 2, that is, positive n for which A108352(n) = 2.

A108373

Numbers whose primal code characteristic = 3, that is, positive n for which A108352(n) = 3.

A108374

Numbers whose primal code characteristic = 4, that is, positive n for which A108352(n) = 4.

TeX Array

$\begin{array}{rcccc||l} 756 & = & 1\!:\!2 & 2\!:\!3 & 4\!:\!1 & 4 \mapsto 1 \mapsto 2 \mapsto 3 \\ 1176 & = & 1\!:\!3 & 2\!:\!1 & 4\!:\!2 & 4 \mapsto 2 \mapsto 1 \mapsto 3 \\ 1188 & = & 1\!:\!2 & 2\!:\!3 & 5\!:\!1 & 5 \mapsto 1 \mapsto 2 \mapsto 3 \\ 1200 & = & 1\!:\!4 & 2\!:\!1 & 3\!:\!2 & 3 \mapsto 2 \mapsto 1 \mapsto 4 \\ 1400 & = & 1\!:\!3 & 3\!:\!2 & 4\!:\!1 & 4 \mapsto 1 \mapsto 3 \mapsto 2 \\ 1404 & = & 1\!:\!2 & 2\!:\!3 & 6\!:\!1 & 6 \mapsto 1 \mapsto 2 \mapsto 3 \\ 1620 & = & 1\!:\!2 & 2\!:\!4 & 3\!:\!1 & 3 \mapsto 1 \mapsto 2 \mapsto 4 \\ 1836 & = & 1\!:\!2 & 2\!:\!3 & 7\!:\!1 & 7 \mapsto 1 \mapsto 2 \mapsto 3 \\ 2052 & = & 1\!:\!2 & 2\!:\!3 & 8\!:\!1 & 8 \mapsto 1 \mapsto 2 \mapsto 3 \\ 2160 & = & 1\!:\!4 & 2\!:\!3 & 3\!:\!1 & 2 \mapsto 3 \mapsto 1 \mapsto 4 \\ 2200 & = & 1\!:\!3 & 3\!:\!2 & 5\!:\!1 & 5 \mapsto 1 \mapsto 3 \mapsto 2 \\ 2400 & = & 1\!:\!5 & 2\!:\!1 & 3\!:\!2 & 3 \mapsto 2 \mapsto 1 \mapsto 5 \\ 2484 & = & 1\!:\!2 & 2\!:\!3 & 9\!:\!1 & 9 \mapsto 1 \mapsto 2 \mapsto 3 \\ 2600 & = & 1\!:\!3 & 3\!:\!2 & 6\!:\!1 & 6 \mapsto 1 \mapsto 3 \mapsto 2 \\ 2904 & = & 1\!:\!3 & 2\!:\!1 & 5\!:\!2 & 5 \mapsto 2 \mapsto 1 \mapsto 3 \end{array}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have the following table:
    * Primal Functions and Functional Digraphs for a(1) to a(15)
    * 0756 = 1:2 2:3 4:1 || 4 -> 1 -> 2 -> 3
    * 1176 = 1:3 2:1 4:2 || 4 -> 2 -> 1 -> 3
    * 1188 = 1:2 2:3 5:1 || 5 -> 1 -> 2 -> 3
    * 1200 = 1:4 2:1 3:2 || 3 -> 2 -> 1 -> 4
    * 1400 = 1:3 3:2 4:1 || 4 -> 1 -> 3 -> 2
    * 1404 = 1:2 2:3 6:1 || 6 -> 1 -> 2 -> 3
    * 1620 = 1:2 2:4 3:1 || 3 -> 1 -> 2 -> 4
    * 1836 = 1:2 2:3 7:1 || 7 -> 1 -> 2 -> 3
    * 2052 = 1:2 2:3 8:1 || 8 -> 1 -> 2 -> 3
    * 2160 = 1:4 2:3 3:1 || 2 -> 3 -> 1 -> 4
    * 2200 = 1:3 3:2 5:1 || 5 -> 1 -> 3 -> 2
    * 2400 = 1:5 2:1 3:2 || 3 -> 2 -> 1 -> 5
    * 2484 = 1:2 2:3 9:1 || 9 -> 1 -> 2 -> 3
    * 2600 = 1:3 3:2 6:1 || 6 -> 1 -> 3 -> 2
    * 2904 = 1:3 2:1 5:2 || 5 -> 2 -> 1 -> 3

A109297

Primal codes of finite permutations on positive integers.

TeX Array

$\begin{array}{rclll} 1 & = & \varnothing \\ 2 & = & 1\!:\!1 \\ 9 & = & 2\!:\!2 \\ 12 & = & 1\!:\!2 & 2\!:\!1 \\ 18 & = & 1\!:\!1 & 2\!:\!2 \\ 40 & = & 1\!:\!3 & 3\!:\!1 \\ 112 & = & 1\!:\!4 & 4\!:\!1 \\ 125 & = & 3\!:\!3 \\ 250 & = & 1\!:\!1 & 3\!:\!3 \\ 352 & = & 1\!:\!5 & 5\!:\!1 \\ 360 & = & 1\!:\!3 & 2\!:\!2 & 3\!:\!1 \\ 540 & = & 1\!:\!2 & 2\!:\!3 & 3\!:\!1 \\ 600 & = & 1\!:\!3 & 2\!:\!1 & 3\!:\!2 \\ 675 & = & 2\!:\!3 & 3\!:\!2 \\ 832 & = & 1\!:\!6 & 6\!:\!1 \\ 1008 & = & 1\!:\!4 & 2\!:\!2 & 4\!:\!1 \\ 1125 & = & 2\!:\!2 & 3\!:\!3 \\ 1350 & = & 1\!:\!1 & 2\!:\!3 & 3\!:\!2 \\ 1500 & = & 1\!:\!2 & 2\!:\!1 & 3\!:\!3 \\ 2176 & = & 1\!:\!7 & 7\!:\!1 \\ 2250 & = & 1\!:\!1 & 2\!:\!2 & 3\!:\!3 \end{array}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have the following table:
    * Primal Codes of Finite Permutations on Positive Integers
    *       1 = { }
    *       2 = 1:1
    *       9 = 2:2
    *      12 = 1:2 2:1
    *      18 = 1:1 2:2
    *      40 = 1:3 3:1
    *     112 = 1:4 4:1
    *     125 = 3:3
    *     250 = 1:1 3:3
    *     352 = 1:5 5:1
    *     360 = 1:3 2:2 3:1
    *     540 = 1:2 2:3 3:1
    *     600 = 1:3 2:1 3:2
    *     675 = 2:3 3:2
    *     832 = 1:6 6:1
    *    1008 = 1:4 2:2 4:1
    *    1125 = 2:2 3:3
    *    1350 = 1:1 2:3 3:2
    *    1500 = 1:2 2:1 3:3
    *    2176 = 1:7 7:1
    *    2250 = 1:1 2:2 3:3

A109298

Primal codes of finite idempotent functions on positive integers.

TeX Array

$\begin{array}{rcllll} 1 & = & \varnothing \\ 2 & = & 1\!:\!1 \\ 9 & = & & 2\!:\!2 \\ 18 & = & 1\!:\!1 & 2\!:\!2 \\ 125 & = & & & 3\!:\!3 \\ 250 & = & 1\!:\!1 & & 3\!:\!3 \\ 1125 & = & & 2\!:\!2 & 3\!:\!3 \\ 2250 & = & 1\!:\!1 & 2\!:\!2 & 3\!:\!3 \\ 2401 & = & & & & 4\!:\!4 \\ 4802 & = & 1\!:\!1 & & & 4\!:\!4 \\ 21609 & = & & 2\!:\!2 & & 4\!:\!4 \\ 43218 & = & 1\!:\!1 & 2\!:\!2 & & 4\!:\!4 \\ 300125 & = & & & 3\!:\!3 & 4\!:\!4 \\ 600250 & = & 1\!:\!1 & & 3\!:\!3 & 4\!:\!4 \\ 2701125 & = & & 2\!:\!2 & 3\!:\!3 & 4\!:\!4 \\ 5402250 & = & 1\!:\!1 & 2\!:\!2 & 3\!:\!3 & 4\!:\!4 \end{array}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have the following table of examples:
    * Primal Codes of Finite Idempotent Functions on Positive Integers
    *       1 = { }
    *       2 = 1:1
    *       9 =     2:2
    *      18 = 1:1 2:2
    *     125 =         3:3
    *     250 = 1:1     3:3
    *    1125 =     2:2 3:3
    *    2250 = 1:1 2:2 3:3
    *    2401 =             4:4
    *    4802 = 1:1         4:4
    *   21609 =     2:2     4:4
    *   43218 = 1:1 2:2     4:4
    *  300125 =         3:3 4:4
    *  600250 = 1:1     3:3 4:4
    * 2701125 =     2:2 3:3 4:4
    * 5402250 = 1:1 2:2 3:3 4:4

A109299

Primal codes of canonical finite permutations on positive integers.

TeX Array

$\begin{array}{rcllll} 1 & = & \varnothing \\ 2 & = & 1\!:\!1 \\ 12 & = & 1\!:\!2 & 2\!:\!1 \\ 18 & = & 1\!:\!1 & 2\!:\!2 \\ 360 & = & 1\!:\!3 & 2\!:\!2 & 3\!:\!1 \\ 540 & = & 1\!:\!2 & 2\!:\!3 & 3\!:\!1 \\ 600 & = & 1\!:\!3 & 2\!:\!1 & 3\!:\!2 \\ 1350 & = & 1\!:\!1 & 2\!:\!3 & 3\!:\!2 \\ 1500 & = & 1\!:\!2 & 2\!:\!1 & 3\!:\!3 \\ 2250 & = & 1\!:\!1 & 2\!:\!2 & 3\!:\!3 \\ 75600 & = & 1\!:\!4 & 2\!:\!3 & 3\!:\!2 & 4\!:\!1 \\ 992250 & = & 1\!:\!1 & 2\!:\!4 & 3\!:\!3 & 4\!:\!2 \\ 105840 & = & 1\!:\!4 & 2\!:\!3 & 3\!:\!1 & 4\!:\!2 \\ 113400 & = & 1\!:\!3 & 2\!:\!4 & 3\!:\!2 & 4\!:\!1 \\ 126000 & = & 1\!:\!4 & 2\!:\!2 & 3\!:\!3 & 4\!:\!1 \\ 158760 & = & 1\!:\!3 & 2\!:\!4 & 3\!:\!1 & 4\!:\!2 \\ 246960 & = & 1\!:\!4 & 2\!:\!2 & 3\!:\!1 & 4\!:\!3 \\ 283500 & = & 1\!:\!2 & 2\!:\!4 & 3\!:\!3 & 4\!:\!1 \\ 294000 & = & 1\!:\!4 & 2\!:\!1 & 3\!:\!3 & 4\!:\!2 \\ 315000 & = & 1\!:\!3 & 2\!:\!2 & 3\!:\!4 & 4\!:\!1 \\ 411600 & = & 1\!:\!4 & 2\!:\!1 & 3\!:\!2 & 4\!:\!3 \\ 472500 & = & 1\!:\!2 & 2\!:\!3 & 3\!:\!4 & 4\!:\!1 \\ 555660 & = & 1\!:\!2 & 2\!:\!4 & 3\!:\!1 & 4\!:\!3 \\ 735000 & = & 1\!:\!3 & 2\!:\!1 & 3\!:\!4 & 4\!:\!2 \\ 864360 & = & 1\!:\!3 & 2\!:\!2 & 3\!:\!1 & 4\!:\!4 \\ 1296540 & = & 1\!:\!2 & 2\!:\!3 & 3\!:\!1 & 4\!:\!4 \\ 1389150 & = & 1\!:\!1 & 2\!:\!4 & 3\!:\!2 & 4\!:\!3 \\ 1440600 & = & 1\!:\!3 & 2\!:\!1 & 3\!:\!2 & 4\!:\!4 \\ 1653750 & = & 1\!:\!1 & 2\!:\!3 & 3\!:\!4 & 4\!:\!2 \\ 2572500 & = & 1\!:\!2 & 2\!:\!1 & 3\!:\!4 & 4\!:\!3 \\ 3241350 & = & 1\!:\!1 & 2\!:\!3 & 3\!:\!2 & 4\!:\!4 \\ 3601500 & = & 1\!:\!2 & 2\!:\!1 & 3\!:\!3 & 4\!:\!4 \\ 3858750 & = & 1\!:\!1 & 2\!:\!2 & 3\!:\!4 & 4\!:\!3 \\ 5402250 & = & 1\!:\!1 & 2\!:\!2 & 3\!:\!3 & 4\!:\!4 \end{array}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have this table:
    * Primal Codes of Canonical Finite Permutations
    *       1 = { }
    *       2 = 1:1
    *      12 = 1:2 2:1
    *      18 = 1:1 2:2
    *     360 = 1:3 2:2 3:1
    *     540 = 1:2 2:3 3:1
    *     600 = 1:3 2:1 3:2
    *    1350 = 1:1 2:3 3:2
    *    1500 = 1:2 2:1 3:3
    *    2250 = 1:1 2:2 3:3
    *   75600 = 1:4 2:3 3:2 4:1
    *  992250 = 1:1 2:4 3:3 4:2
    *  105840 = 1:4 2:3 3:1 4:2
    *  113400 = 1:3 2:4 3:2 4:1
    *  126000 = 1:4 2:2 3:3 4:1
    *  158760 = 1:3 2:4 3:1 4:2
    *  246960 = 1:4 2:2 3:1 4:3
    *  283500 = 1:2 2:4 3:3 4:1
    *  294000 = 1:4 2:1 3:3 4:2
    *  315000 = 1:3 2:2 3:4 4:1
    *  411600 = 1:4 2:1 3:2 4:3
    *  472500 = 1:2 2:3 3:4 4:1
    *  555660 = 1:2 2:4 3:1 4:3
    *  735000 = 1:3 2:1 3:4 4:2
    *  864360 = 1:3 2:2 3:1 4:4
    * 1296540 = 1:2 2:3 3:1 4:4
    * 1389150 = 1:1 2:4 3:2 4:3
    * 1440600 = 1:3 2:1 3:2 4:4
    * 1653750 = 1:1 2:3 3:4 4:2
    * 2572500 = 1:2 2:1 3:4 4:3
    * 3241350 = 1:1 2:3 3:2 4:4
    * 3601500 = 1:2 2:1 3:3 4:4
    * 3858750 = 1:1 2:2 3:4 4:3
    * 5402250 = 1:1 2:2 3:3 4:4

A109300

a(n) = number of positive integers whose rote height in gammas is n.

JPEG

$\text{Table 1.} ~~ \text{Rotes and Primal Functions for Positive Integers of Rote Height 2}\!$
$\begin{array}{l} 2\!:\!1 \\ 3 \end{array}$	$\begin{array}{l} 1\!:\!2 \\ 4 \end{array}$	$\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}$	$\begin{array}{l} 2\!:\!2 \\ 9 \end{array}$	$\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}$	$\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}$	$\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}$

ASCII

 Example

    * Table of Rotes and Primal Functions for Positive Integers of Rote Height 2
    *                                                                          
    * o-o     o-o       o-o   o-o o-o     o-o o-o       o-o o-o     o-o o-o o-o
    * |       |         |     |   |       |   |         |   |       |   |   |  
    * o-o   o-o     o-o o-o   o---o     o-o   o-o   o-o o---o     o-o   o---o  
    * |     |       |   |     |         |     |     |   |         |     |      
    * O     O       O===O     O         O=====O     O===O         O=====O      
    *                                                                          
    * 2:1   1:2     1:1 2:1   2:2       1:2 2:1     1:1 2:2       1:2 2:2      
    *                                                                          
    * 3     4       6         9         12          18            36           
    *

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.

Example

$802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1$

$\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}$

$\begin{array}{lllll} 802701 & = & 9 \cdot 89189 & = & 2\!:\!2 ~~ 8638\!:\!1 \\ 8638 & = & 2 \cdot 7 \cdot 617 & = & 1\!:\!1 ~~ 4\!:\!1 ~~ 113\!:\!1 \\ 113 & & & = & 30\!:\!1 \\ 30 & = & 2 \cdot 3 \cdot 5 & = & 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 4 & & & = & 1\!:\!2 \\ 3 & & & = & 2\!:\!1 \\ 2 & & & = & 1\!:\!1 \end{array}$

$\text{So the rote of 802701 is the following graph:}\!$

$\text{By inspection, the rote height of 802701 is 6.}\!$

JPEG

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

ASCII

 Comment

    * Table of Rotes and Primal Functions for Positive Integers from 1 to 40
    *                                                                        
    *                                                         o-o            
    *                                                         |              
    *                             o-o             o-o         o-o            
    *                             |               |           |              
    *               o-o           o-o           o-o           o-o            
    *               |             |             |             |              
    * O             O             O             O             O              
    *                                                                        
    * { }           1:1           2:1           1:2           3:1            
    *                                                                        
    * 1             2             3             4             5              
    *                                                                        
    *                                                                        
    *                 o-o           o-o                           o-o        
    *                 |             |                             |          
    *     o-o       o-o             o-o         o-o o-o           o-o        
    *     |         |               |           |   |             |          
    * o-o o-o       o-o           o-o           o---o         o-o o-o        
    * |   |         |             |             |             |   |          
    * O===O         O             O             O             O===O          
    *                                                                        
    * 1:1 2:1       4:1           1:3           2:2           1:1 3:1        
    *                                                                        
    * 6             7             8             9             10             
    *                                                                        
    *                                                                        
    * o-o                                                                    
    * |                                                                      
    * o-o                             o-o             o-o         o-o        
    * |                               |               |           |          
    * o-o             o-o o-o     o-o o-o           o-o       o-o o-o        
    * |               |   |       |   |             |         |   |          
    * o-o           o-o   o-o     o===o-o       o-o o-o       o-o o-o        
    * |             |     |       |             |   |         |   |          
    * O             O=====O       O             O===O         O===O          
    *                                                                        
    * 5:1           1:2 2:1       6:1           1:1 4:1       2:1 3:1        
    *                                                                        
    * 11            12            13            14            15             
    *                                                                        
    *                                                                        
    *                 o-o                         o-o                        
    *                 |                           |                          
    *     o-o       o-o                           o-o               o-o      
    *     |         |                             |                 |        
    *   o-o         o-o               o-o o-o   o-o             o-o o-o      
    *   |           |                 |   |     |               |   |        
    * o-o           o-o           o-o o---o     o-o           o-o   o-o      
    * |             |             |   |         |             |     |        
    * O             O             O===O         O             O=====O        
    *                                                                        
    * 1:4           7:1           1:1 2:2       8:1           1:2 3:1        
    *                                                                        
    * 16            17            18            19            20             
    *                                                                        
    *                                                                        
    *                   o-o                                                  
    *                   |                                                    
    *       o-o         o-o       o-o o-o         o-o         o-o            
    *       |           |         |   |           |           |              
    * o-o o-o           o-o       o---o           o-o o-o     o-o o-o        
    * |   |             |         |               |   |       |   |          
    * o-o o-o       o-o o-o       o-o           o-o   o-o     o---o          
    * |   |         |   |         |             |     |       |              
    * O===O         O===O         O             O=====O       O              
    *                                                                        
    * 2:1 4:1       1:1 5:1       9:1           1:3 2:1       3:2            
    *                                                                        
    * 21            22            23            24            25             
    *                                                                        
    *                                                                        
    *                                               o-o                      
    *                                               |                        
    *         o-o       o-o               o-o       o-o               o-o    
    *         |         |                 |         |                 |      
    *     o-o o-o   o-o o-o         o-o o-o     o-o o-o           o-o o-o    
    *     |   |     |   |           |   |       |   |             |   |      
    * o-o o===o-o   o---o         o-o   o-o     o===o-o       o-o o-o o-o    
    * |   |         |             |     |       |             |   |   |      
    * O===O         O             O=====O       O             O===O===O      
    *                                                                        
    * 1:1 6:1       2:3           1:2 4:1       10:1          1:1 2:1 3:1    
    *                                                                        
    * 26            27            28            29            30             
    *                                                                        
    *                                                                        
    * o-o                                                                    
    * |                                                                      
    * o-o             o-o             o-o             o-o                    
    * |               |               |               |                      
    * o-o             o-o             o-o           o-o       o-o   o-o      
    * |               |               |             |         |     |        
    * o-o             o-o         o-o o-o           o-o       o-o o-o        
    * |               |           |   |             |         |   |          
    * o-o           o-o           o-o o-o       o-o o-o       o-o o-o        
    * |             |             |   |         |   |         |   |          
    * O             O             O===O         O===O         O===O          
    *                                                                        
    * 11:1          1:5           2:1 5:1       1:1 7:1       3:1 4:1        
    *                                                                        
    * 31            32            33            34            35             
    *                                                                        
    *                                                                        
    *                                   o-o                                  
    *                                   |                                    
    *                 o-o o-o           o-o             o-o     o-o o-o      
    *                 |   |             |               |       |   |        
    *   o-o o-o o-o o-o   o-o         o-o       o-o o-o o-o     o-o o-o      
    *   |   |   |   |     |           |         |   |   |       |   |        
    * o-o   o---o   o=====o-o     o-o o-o       o-o o===o-o   o-o   o-o      
    * |     |       |             |   |         |   |         |     |        
    * O=====O       O             O===O         O===O         O=====O        
    *                                                                        
    * 1:2 2:2       12:1          1:1 8:1       2:1 6:1       1:3 3:1        
    *                                                                        
    * 36            37            38            39            40             
    *                                                                        
    * In these Figures, "extended lines of identity" like o===o
    * indicate identified nodes and capital O is the root node.
    * The rote height in gammas is found by finding the number
    * of graphs of the following shape between the root and one
    * of the highest nodes of the tree:
    * o--o
    * |
    * o
    * A sequence like this, that can be regarded as a nonnegative integer
    * measure on positive integers, may have as many as 3 other sequences
    * associated with it. Given that the fiber of a function f at n is all
    * the domain elements that map to n, we always have the fiber minimum
    * or minimum inverse function and may also have the fiber cardinality
    * and the fiber maximum or maximum inverse function. For A109301, the
    * minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the
    * first positive integer whose rote height is n, the fiber cardinality
    * is A109300, giving the number of positive integers of rote height n,
    * while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving
    * the last positive integer whose rote height is n, has the following
    * initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36,
    * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
    * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
    * 7.840858554516122655953405327738 x 10^371.

 Example

    * Writing (prime(i))^j as i:j, we have:
    * 802701 = 2:2 8638:1
    * 8638 = 1:1 4:1 113:1
    * 113 = 30:1
    * 30 = 1:1 2:1 3:1
    * 4 = 1:2
    * 3 = 2:1
    * 2 = 1:1
    * 1 = { }
    * So rote(802701) is the graph:
    *                              
    *                           o-o
    *                           |  
    *                       o-o o-o
    *                       |   |  
    *               o-o o-o o-o o-o
    *               |   |   |   |  
    *             o-o   o===o===o-o
    *             |     |          
    * o-o o-o o-o o-o   o---------o
    * |   |   |   |     |          
    * o---o   o===o=====o---------o
    * |       |                    
    * O=======O                    
    *                              
    * Therefore rhig(802701) = 6.

A111788

Order of the domain D_n (n >= 0) in the inverse limit domain D_infinity.

A111789

First differences of (0, A111788), the sequence that begins with 0 and continues with the terms of A111788.

A111790

Partial sums of A111788.

A111791

Positive integers sorted by rote height, as measured by A109301.

TeX Array

$\begin{array}{l|l|r} h & S_h ~=~ \{ m ~:~ \operatorname{rote~height}(m) ~=~ \operatorname{A109301}(m) ~=~ h \} & |S_h| \\\hline\hline 0 & \{ 1 \} & 1 \\\hline 1 & \{ 2 \} & 1 \\\hline 2 & \{ 3, 4, 6, 9, 12, 18, 36 \} & 7 \\\hline 3 & \{ 5, 7, 8, 10, 13, 14, 15, 16, 20, 21, 23, 24, 25, 26, 27, 28, 30, \\ & 35, 37, 39, 40, 42, 45, 46, 48, 49, 50, 52, 54, 56, 60, 61, 63, \\ & 64, 65, 69, 70, 72, 74, 75, 78, 80, 81, 84, 90, 91, 92, 98, 100, \ldots \} & 999999991 \\\hline 4 & \{ 11, 17, 19, 22, 29, 32, 33, 34, 38, 41, 43, 44, 47, 51, 53, 55, \\ & 57, 58, 66, 68, 71, 73, 76, 77, 82, 83, 85, 86, 87, 88, 89, 94, \\ & 95, 96, 97, 99, \ldots \} & \operatorname{A109300}(4) \\\hline 5 & \{ 31, 59, 62, 67, 79, 93, \ldots \} & \operatorname{A109300}(5) \end{array}$

Wiki Table

$\text{Table in which the}~ h^{\text{th}} ~\text{row lists the positive integers of rote height}~ h$
h	m such that rhig(m) = A109301(m) = h
0	1
1	2
2	3 4 6 9 12 18 36
3	5 7 8 10 13 14 15 16 20 21 23 24 25 26 27 28 30 35 37 39 40 42 45 46 48 49 50 52 54 56 60 61 63 64 65 69 70 72 74 75 78 80 81 84 90 91 92 98 100 …
4	11 17 19 22 29 32 33 34 38 41 43 44 47 51 53 55 57 58 66 68 71 73 76 77 82 83 85 86 87 88 89 94 95 96 97 99 …
5	31 59 62 67 79 93 …
Smallest m in the h^th row = A007097. Number of values in the h^th row = A109300(h). Number of values up through the h^th row = A050924(h + 1).

ASCII

 Example

    * Table in which the h^th row lists the positive integers of rote height h:
    * h | m such that rhig(m) = A109301(m) = h
    * --+------------------------------------------------------
    * 0 | 1
    * --+------------------------------------------------------
    * 1 | 2
    * --+------------------------------------------------------
    * 2 | 3 4 6 9 12 18 36
    * --+------------------------------------------------------
    * 3 | 5 7 8 10 13 14 15 16 20 21 23 24 25 26 27 28 30
    *   | 35 37 39 40 42 45 46 48 49 50 52 54 56 60 61 63
    *   | 64 65 69 70 72 74 75 78 80 81 84 90 91 92 98 100 ...
    * --+------------------------------------------------------
    * 4 | 11 17 19 22 29 32 33 34 38 41 43 44 47 51 53 55
    *   | 57 58 66 68 71 73 76 77 82 83 85 86 87 88 89 94
    *   | 95 96 97 99 ...
    * --+------------------------------------------------------
    * 5 | 31 59 62 67 79 93 ...
    * --+------------------------------------------------------
    * First column = A007097. Count in h^th row = A109300(h).
    * Cumulative count up through the h^th row = A050924(h+1).

A111792

Positive integers sorted by rote weight (A062537) and rote height (A109301).

TeX Array

$\begin{array}{rcl|rr|r|r} a & & \operatorname{code} & g & h & s & t \\ \hline 1 & = & \varnothing & 0 & 0 & 1 & 1 \\ \hline 2 & = & 1\!:\!1 & 1 & 1 & 1 & 1 \\ \hline 3 & = & 2\!:\!1 & 2 & 2 & & \\ 4 & = & 1\!:\!2 & 2 & 2 & 2 & 2 \\ \hline 6 & = & 1\!:\!1 ~~ 2\!:\!1 & 3 & 2 & & \\ 9 & = & 2\!:\!2 & 3 & 2 & 2 & \\ \hline 5 & = & 3\!:\!1 & 3 & 3 & & \\ 7 & = & 4\!:\!1 & 3 & 3 & & \\ 8 & = & 1\!:\!3 & 3 & 3 & & \\ 16 & = & 1\!:\!4 & 3 & 3 & 4 & 6 \\ \hline 12 & = & 1\!:\!2 ~~ 2\!:\!1 & 4 & 2 & & \\ 18 & = & 1\!:\!1 ~~ 2\!:\!2 & 4 & 2 & 2 & \\ \hline 10 & = & 1\!:\!1 ~~ 3\!:\!1 & 4 & 3 & & \\ 13 & = & 6\!:\!1 & 4 & 3 & & \\ 14 & = & 1\!:\!1 ~~ 4\!:\!1 & 4 & 3 & & \\ 23 & = & 9\!:\!1 & 4 & 3 & & \\ 25 & = & 3\!:\!2 & 4 & 3 & & \\ 27 & = & 2\!:\!3 & 4 & 3 & & \\ 49 & = & 4\!:\!2 & 4 & 3 & & \\ 64 & = & 1\!:\!6 & 4 & 3 & & \\ 81 & = & 2\!:\!4 & 4 & 3 & & \\ 512 & = & 1\!:\!9 & 4 & 3 & 10 & \\ \hline 11 & = & 5\!:\!1 & 4 & 4 & & \\ 17 & = & 7\!:\!1 & 4 & 4 & & \\ 19 & = & 8\!:\!1 & 4 & 4 & & \\ 32 & = & 1\!:\!5 & 4 & 4 & & \\ 53 & = & 16\!:\!1 & 4 & 4 & & \\ 128 & = & 1\!:\!7 & 4 & 4 & & \\ 256 & = & 1\!:\!8 & 4 & 4 & & \\ 65536 & = & 1\!:\!16 & 4 & 4 & 8 & 20 \\ \hline \end{array}$

ASCII

 Example

    * Table of Integers, Primal Codes, Sort Parameters and Subtotals
    *     a   code    | g h | s | t
    * ----------------+-----+---+---
    *     1 = { }     | 0 0 | 1 | 1
    * ----------------+-----+---+---
    *     2 = 1:1     | 1 1 | 1 | 1
    * ----------------+-----+---+---
    *     3 = 2:1     | 2 2 |   |
    *     4 = 1:2     | 2 2 | 2 | 2
    * ----------------+-----+---+---
    *     6 = 1:1 2:1 | 3 2 |   |
    *     9 = 2:2     | 3 2 | 2 |
    * ----------------+-----+---+---
    *     5 = 3:1     | 3 3 |   |
    *     7 = 4:1     | 3 3 |   |
    *     8 = 1:3     | 3 3 |   |
    *    16 = 1:4     | 3 3 | 4 | 6
    * ----------------+-----+---+---
    *    12 = 1:2 2:1 | 4 2 |   |
    *    18 = 1:1 2:2 | 4 2 | 2 |
    * ----------------+-----+---+---
    *    10 = 1:1 3:1 | 4 3 |   |
    *    13 = 6:1     | 4 3 |   |
    *    14 = 1:1 4:1 | 4 3 |   |
    *    23 = 9:1     | 4 3 |   |
    *    25 = 3:2     | 4 3 |   |
    *    27 = 2:3     | 4 3 |   |
    *    49 = 4:2     | 4 3 |   |
    *    64 = 1:6     | 4 3 |   |
    *    81 = 2:4     | 4 3 |   |
    *   512 = 1:9     | 4 3 |10 |
    * ----------------+-----+---+---
    *    11 = 5:1     | 4 4 |   |
    *    17 = 7:1     | 4 4 |   |
    *    19 = 8:1     | 4 4 |   |
    *    32 = 1:5     | 4 4 |   |
    *    53 = 16:1    | 4 4 |   |
    *   128 = 1:7     | 4 4 |   |
    *   256 = 1:8     | 4 4 |   |
    * 65536 = 1:16    | 4 4 | 8 |20
    * ----------------+-----+---+---
    * a = this sequence
    * g = rote weight in gammas = A062537
    * h = rote height in gammas = A109301
    * s = count in (g, h) class = A111793
    * t = count in weight class = A061396

A111793

Triangle T(g, h) = number of rotes of weight g and height h, both in gammas.

TeX Array

$\begin{array}{l|rrrrrr} g \backslash h & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 1 \\ 1 & & 1 \\ 2 & & & 2 \\ 3 & & & 2 & 4 \\ 4 & & & 2 & 10 & 8 \\ 5 & & & 1 & 24 & 32 & 16 \end{array}$

ASCII

 Example

    * Table T(g, h), omitting zeros, starts out as follows:
    * g\h| 0   1   2   3   4   5
    * ---+-----------------------
    *  0 | 1
    *  1 |     1
    *  2 |         2
    *  3 |         2   4
    *  4 |         2  10   8
    *  5 |         1  24  32  16

A111794

Integers whose rote weight and rote height are equal, sorted by the equated value.

TeX Array

$\begin{array}{l|l|r} j & S_j ~=~ \{ m ~:~ \operatorname{g}(m) ~=~ \operatorname{h}(m) ~=~ j \} & |S_j| \\\hline\hline 0 & \{ 1 \} & 1 \\\hline 1 & \{ 2 \} & 1 \\\hline 2 & \{ 3, 4 \} & 2 \\\hline 3 & \{ 5, 7, 8, 16 \} & 4 \\\hline 4 & \{ 11, 17, 19, 32, 53, 128, 256, 65536 \} & 8 \\\hline 5 & \{ 31, 59, 67, 131, 241, 719, 1619, 2048, 131072, 524288, 821641, \\ & 4294967296, 9007199254740992, 2^{128}, 2^{256}, 2^{65536} \} & 16 \end{array}$

Wiki Table

$\text{Table whose}~ j^{\text{th}} ~\text{row lists the integers}~ m ~\text{with}~ g(m) = h(m) = j$
j	m such that g(m) = h(m) = j
0	1
1	2
2	3 4
3	5 7 8 16
4	11 17 19 32 53 128 256 65536
5	31 59 67 131 241 719 1619 2048 131072 524288 821641 4294967296 9007199254740992 2¹²⁸ 2²⁵⁶ 2⁶⁵⁵³⁶

ASCII

 Example

    * Triangle whose j^th row lists the integers m with g(m) = h(m) = j
    * j | m such that g(m) = h(m) = j
    * --+-------------------------------------------------------
    * 0 | 1
    * 1 | 2
    * 2 | 3 4
    * 3 | 5 7 8 16
    * 4 | 11 17 19 32 53 128 256 65536
    * 5 | 31 59 67 131 241 719 1619 2048 131072 524288 821641
    *   | 4294967296 9007199254740992 2^128 2^256 2^65536

A111795

Positive integers whose rote weight and rote height are equal.

JPEG

$\begin{array}{l} \varnothing \\ 1 \end{array}$

$\begin{array}{l} 1\!:\!1 \\ 2 \end{array}$

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

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

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

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

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

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

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

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

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

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

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

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

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

ASCII

 Example

    * Tables of Rotes and Primal Codes for a(1) to a(9)
    *                                                              
    *                                                 o-o          
    *                                                 |            
    *                           o-o     o-o     o-o   o-o       o-o
    *                           |       |       |     |         |  
    *             o-o     o-o   o-o   o-o       o-o   o-o     o-o  
    *             |       |     |     |         |     |       |    
    *       o-o   o-o   o-o     o-o   o-o     o-o     o-o   o-o    
    *       |     |     |       |     |       |       |     |      
    * O     O     O     O       O     O       O       O     O      
    *                                                              
    * { }   1:1   2:1   1:2     3:1   4:1     1:3     5:1   1:4    
    *                                                              
    * 1     2     3     4       5     7       8       11    16     
    *

A111796

Positive integers sorted by rote weight (A062537) and omega (A001221).

TeX Array

$\begin{array}{rcl|rr|r|r} a & & \operatorname{code} & g & w & s & t \\ \hline 1 & = & \varnothing & 0 & 0 & 1 & 1 \\ \hline 2 & = & 1\!:\!1 & 1 & 1 & 1 & 1 \\ \hline 3 & = & 2\!:\!1 & 2 & 1 & & \\ 4 & = & 1\!:\!2 & 2 & 1 & 2 & 2 \\ \hline 5 & = & 3\!:\!1 & 3 & 1 & & \\ 7 & = & 4\!:\!1 & 3 & 1 & & \\ 8 & = & 1\!:\!3 & 3 & 1 & & \\ 9 & = & 2\!:\!2 & 3 & 1 & & \\ 16 & = & 1\!:\!4 & 3 & 1 & 5 & \\ \hline 6 & = & 1\!:\!1 ~~ 2\!:\!1 & 3 & 2 & 1 & 6 \\ \hline 11 & = & 5\!:\!1 & 4 & 1 & & \\ 13 & = & 6\!:\!1 & 4 & 1 & & \\ 17 & = & 7\!:\!1 & 4 & 1 & & \\ 19 & = & 8\!:\!1 & 4 & 1 & & \\ 23 & = & 9\!:\!1 & 4 & 1 & & \\ 25 & = & 3\!:\!2 & 4 & 1 & & \\ 27 & = & 2\!:\!3 & 4 & 1 & & \\ 32 & = & 1\!:\!5 & 4 & 1 & & \\ 49 & = & 4\!:\!2 & 4 & 1 & & \\ 53 & = & 16\!:\!1 & 4 & 1 & & \\ 64 & = & 1\!:\!6 & 4 & 1 & & \\ 81 & = & 2\!:\!4 & 4 & 1 & & \\ 128 & = & 1\!:\!7 & 4 & 1 & & \\ 256 & = & 1\!:\!8 & 4 & 1 & & \\ 512 & = & 1\!:\!9 & 4 & 1 & & \\ 65536 & = & 1\!:\!16 & 4 & 1 & 16 & \\ \hline 10 & = & 1\!:\!1 ~~ 3\!:\!1 & 4 & 2 & & \\ 12 & = & 1\!:\!2 ~~ 2\!:\!1 & 4 & 2 & & \\ 14 & = & 1\!:\!1 ~~ 4\!:\!1 & 4 & 2 & & \\ 18 & = & 1\!:\!1 ~~ 2\!:\!2 & 4 & 2 & 4 & 20 \\ \hline \end{array}$

ASCIII

 Example

    * Table of Integers, Primal Codes, Sort Parameters and Subtotals
    *     a   code    | g w | s | t
    * ----------------+-----+---+---
    *     1 = { }     | 0 0 | 1 | 1
    * ----------------+-----+---+---
    *     2 = 1:1     | 1 1 | 1 | 1
    * ----------------+-----+---+---
    *     3 = 2:1     | 2 1 |   |
    *     4 = 1:2     | 2 1 | 2 | 2
    * ----------------+-----+---+---
    *     5 = 3:1     | 3 1 |   |
    *     7 = 4:1     | 3 1 |   |
    *     8 = 1:3     | 3 1 |   |
    *     9 = 2:2     | 3 1 |   |
    *    16 = 1:4     | 3 1 | 5 |
    * ----------------+-----+---+---
    *     6 = 1:1 2:1 | 3 2 | 1 | 6
    * ----------------+-----+---+---
    *    11 = 5:1     | 4 1 |   |
    *    13 = 6:1     | 4 1 |   |
    *    17 = 7:1     | 4 1 |   |
    *    19 = 8:1     | 4 1 |   |
    *    23 = 9:1     | 4 1 |   |
    *    25 = 3:2     | 4 1 |   |
    *    27 = 2:3     | 4 1 |   |
    *    32 = 1:5     | 4 1 |   |
    *    49 = 4:2     | 4 1 |   |
    *    53 = 16:1    | 4 1 |   |
    *    64 = 1:6     | 4 1 |   |
    *    81 = 2:4     | 4 1 |   |
    *   128 = 1:7     | 4 1 |   |
    *   256 = 1:8     | 4 1 |   |
    *   512 = 1:9     | 4 1 |   |
    * 65536 = 1:16    | 4 1 |16 |
    * ----------------+-----+---+---
    *    10 = 1:1 3:1 | 4 2 |   |
    *    12 = 1:2 2:1 | 4 2 |   |
    *    14 = 1:1 4:1 | 4 2 |   |
    *    18 = 1:1 2:2 | 4 2 | 4 |20
    * ----------------+-----+---+---
    * a = this sequence
    * g = rote weight in gammas = A062537
    * w = rote wayage in gammas = A001221
    * s = count in (g, w) class = A111797
    * t = count in weight class = A061396

A111797

Triangle T(g, w) = number of rotes of weight g and wayage w.

TeX Array

$\begin{array}{l|rrrrrr} g \backslash w & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 1 \\ 1 & & 1 \\ 2 & & 2 \\ 3 & & 5 & 1 \\ 4 & & 16 & 4 \\ 5 & & 56 & 17 \end{array}$

ASCII

 Example

    * Table T(g, w), omitting zeros, begins as follows:
    * g\w| 0   1   2   3   4   5
    * ---+-----------------------
    *  0 | 1
    *  1 |     1
    *  2 |     2
    *  3 |     5   1
    *  4 |    16   4
    *  5 |    56  17

A111798

Positive integers sorted by rote height (A109301) and omega (A001221).

TeX Array

$\begin{array}{l|r|rr|r|r} \text{Primal Function} & \text{Primal Code} ~=~ a & h & w & s & t \\ \hline \varnothing & 1 & 0 & 0 & 1 & 1 \\ \hline 1\!:\!1 & 2 & 1 & 1 & 1 & 1 \\ \hline 2\!:\!1 & 3 & 2 & 1 & & \\ 1\!:\!2 & 4 & 2 & 1 & & \\ 2\!:\!2 & 9 & 2 & 1 & 3 & \\ \hline 1\!:\!1 ~~ 2\!:\!1 & 6 & 2 & 2 & & \\ 1\!:\!2 ~~ 2\!:\!1 & 12 & 2 & 2 & & \\ 1\!:\!1 ~~ 2\!:\!2 & 18 & 2 & 2 & & \\ 1\!:\!2 ~~ 2\!:\!2 & 36 & 2 & 2 & 4 & 7 \\ \hline & & & & & \\ 1\!:\!3 & 8 & 3 & 1 & & \\ 1\!:\!4 & 16 & 3 & 1 & & \\ 1\!:\!6 & 64 & 3 & 1 & & \\ 1\!:\!9 & 512 & 3 & 1 & & \\ 1\!:\!12 & 4096 & 3 & 1 & & \\ 1\!:\!18 & 262144 & 3 & 1 & & \\ 1\!:\!36 & 68719476736 & 3 & 1 & & \\ & & & & & \\ 2\!:\!3 & 27 & 3 & 1 & & \\ 2\!:\!4 & 81 & 3 & 1 & & \\ 2\!:\!6 & 729 & 3 & 1 & & \\ 2\!:\!9 & 19683 & 3 & 1 & & \\ 2\!:\!12 & 531441 & 3 & 1 & & \\ 2\!:\!18 & 387420489 & 3 & 1 & & \\ 2\!:\!36 & 150094635296999121 & 3 & 1 & & \\ & & & & & \\ 3\!:\!1 & 5 & 3 & 1 & & \\ 4\!:\!1 & 7 & 3 & 1 & & \\ 6\!:\!1 & 13 & 3 & 1 & & \\ 9\!:\!1 & 23 & 3 & 1 & & \\ 12\!:\!1 & 37 & 3 & 1 & & \\ 18\!:\!1 & 61 & 3 & 1 & & \\ 36\!:\!1 & 151 & 3 & 1 & & \\ & & & & & \\ 3\!:\!2 & 25 & 3 & 1 & & \\ 4\!:\!2 & 49 & 3 & 1 & & \\ 6\!:\!2 & 169 & 3 & 1 & & \\ 9\!:\!2 & 529 & 3 & 1 & & \\ 12\!:\!2 & 1369 & 3 & 1 & & \\ 18\!:\!2 & 3721 & 3 & 1 & & \\ 36\!:\!2 & 22801 & 3 & 1 & & \\ & & & & & \\ 3\!:\!3 & 125 & 3 & 1 & & \\ 3\!:\!4 & 625 & 3 & 1 & & \\ 3\!:\!6 & 15625 & 3 & 1 & & \\ 3\!:\!9 & 1953125 & 3 & 1 & & \\ 3\!:\!12 & 244140625 & 3 & 1 & & \\ 3\!:\!18 & 3814697265625 & 3 & 1 & & \\ 3\!:\!36 & 14551915228366851806640625 & 3 & 1 & & \\ & & & & & \\ 4\!:\!3 & 343 & 3 & 1 & & \\ 4\!:\!4 & 2401 & 3 & 1 & & \\ 4\!:\!6 & 117649 & 3 & 1 & & \\ 4\!:\!9 & 40353607 & 3 & 1 & & \\ 4\!:\!12 & 13841287201 & 3 & 1 & & \\ 4\!:\!18 & 1628413597910449 & 3 & 1 & & \\ 4\!:\!36 & 2651730845859653471779023381601 & 3 & 1 & & \\ & & & & & \\ 6\!:\!3 & 2197 & 3 & 1 & & \\ 6\!:\!4 & 28561 & 3 & 1 & & \\ 6\!:\!6 & 4826809 & 3 & 1 & & \\ 6\!:\!9 & 10604499373 & 3 & 1 & & \\ 6\!:\!12 & 23298085122481 & 3 & 1 & & \\ 6\!:\!18 & 112455406951957393129 & 3 & 1 & & \\ 6\!:\!36 & 13^{36} & 3 & 1 & & \\ & & & & & \\ 9\!:\!3 & 12167 & 3 & 1 & & \\ 9\!:\!4 & 279841 & 3 & 1 & & \\ 9\!:\!6 & 148035889 & 3 & 1 & & \\ 9\!:\!9 & 1801152661463 & 3 & 1 & & \\ 9\!:\!12 & 21914624432020321 & 3 & 1 & & \\ 9\!:\!18 & 3244150909895248285300369 & 3 & 1 & & \\ 9\!:\!36 & 23^{36} & 3 & 1 & & \\ & & & & & \\ 12\!:\!3 & 50653 & 3 & 1 & & \\ 12\!:\!4 & 1874161 & 3 & 1 & & \\ 12\!:\!6 & 2565726409 & 3 & 1 & & \\ 12\!:\!9 & 129961739795077 & 3 & 1 & & \\ 12\!:\!12 & 6582952005840035281 & 3 & 1 & & \\ 12\!:\!18 & 16890053810563300749953435929 & 3 & 1 & & \\ 12\!:\!36 & 37^{36} & 3 & 1 & & \\ & & & & & \\ 18\!:\!3 & 226981 & 3 & 1 & & \\ 18\!:\!4 & 13845841 & 3 & 1 & & \\ 18\!:\!6 & 51520374361 & 3 & 1 & & \\ 18\!:\!9 & 11694146092834141 & 3 & 1 & & \\ 18\!:\!12 & 2654348974297586158321 & 3 & 1 & & \\ 18\!:\!18 & 136753052840548005895349735207881 & 3 & 1 & & \\ 18\!:\!36 & 61^{36} & 3 & 1 & & \\ & & & & & \\ 36\!:\!3 & 3442951 & 3 & 1 & & \\ 36\!:\!4 & 519885601 & 3 & 1 & & \\ 36\!:\!6 & 11853911588401 & 3 & 1 & & \\ 36\!:\!9 & 40812436757196811351 & 3 & 1 & & \\ 36\!:\!12 & 140515219945627518837736801 & 3 & 1 & & \\ 36\!:\!18 & 151^{18} & 3 & 1 & & \\ 36\!:\!36 & 151^{36} & 3 & 1 & 77 & \\ \hline \end{array}$

ASCII

 Example

    * Table of Primal Functions, Codes, Sort Parameters and Subtotals
    * Primal Function |           Primal Code     =     a | h w | s | t
    * ----------------+-----------------------------------+-----+---+---
    * { }             |                                 1 | 0 0 | 1 | 1
    * ----------------+-----------------------------------+-----+---+---
    * 1:1             |                                 2 | 1 1 | 1 | 1
    * ----------------+-----------------------------------+-----+---+---
    * 2:1             |                                 3 | 2 1 |   |
    * 1:2             |                                 4 | 2 1 |   |
    * 2:2             |                                 9 | 2 1 | 3 |
    * ----------------+-----------------------------------+-----+---+---
    * 1:1 2:1         |                                 6 | 2 2 |   |
    * 1:2 2:1         |                                12 | 2 2 |   |
    * 1:1 2:2         |                                18 | 2 2 |   |
    * 1:2 2:2         |                                36 | 2 2 | 4 | 7
    * ----------------+-----------------------------------+-----+---+---
    *                 |                                   |     |   |
    * 1:3             |                                 8 | 3 1 |   |
    * 1:4             |                                16 | 3 1 |   |
    * 1:6             |                                64 | 3 1 |   |
    * 1:9             |                               512 | 3 1 |   |
    * 1:12            |                              4096 | 3 1 |   |
    * 1:18            |                            262144 | 3 1 |   |
    * 1:36            |                       68719476736 | 3 1 |   |
    *                 |                                   |     |   |
    * 2:3             |                                27 | 3 1 |   |
    * 2:4             |                                81 | 3 1 |   |
    * 2:6             |                               729 | 3 1 |   |
    * 2:9             |                             19683 | 3 1 |   |
    * 2:12            |                            531441 | 3 1 |   |
    * 2:18            |                         387420489 | 3 1 |   |
    * 2:36            |                150094635296999121 | 3 1 |   |
    *                 |                                   |     |   |
    * 3:1             |                                 5 | 3 1 |   |
    * 4:1             |                                 7 | 3 1 |   |
    * 6:1             |                                13 | 3 1 |   |
    * 9:1             |                                23 | 3 1 |   |
    * 12:1            |                                37 | 3 1 |   |
    * 18:1            |                                61 | 3 1 |   |
    * 36:1            |                               151 | 3 1 |   |
    *                 |                                   |     |   |
    * 3:2             |                                25 | 3 1 |   |
    * 4:2             |                                49 | 3 1 |   |
    * 6:2             |                               169 | 3 1 |   |
    * 9:2             |                               529 | 3 1 |   |
    * 12:2            |                              1369 | 3 1 |   |
    * 18:2            |                              3721 | 3 1 |   |
    * 36:2            |                             22801 | 3 1 |   |
    *                 |                                   |     |   |
    * 3:3             |                               125 | 3 1 |   |
    * 3:4             |                               625 | 3 1 |   |
    * 3:6             |                             15625 | 3 1 |   |
    * 3:9             |                           1953125 | 3 1 |   |
    * 3:12            |                         244140625 | 3 1 |   |
    * 3:18            |                     3814697265625 | 3 1 |   |
    * 3:36            |        14551915228366851806640625 | 3 1 |   |
    *                 |                                   |     |   |
    * 4:3             |                               343 | 3 1 |   |
    * 4:4             |                              2401 | 3 1 |   |
    * 4:6             |                            117649 | 3 1 |   |
    * 4:9             |                          40353607 | 3 1 |   |
    * 4:12            |                       13841287201 | 3 1 |   |
    * 4:18            |                  1628413597910449 | 3 1 |   |
    * 4:36            |   2651730845859653471779023381601 | 3 1 |   |
    *                 |                                   |     |   |
    * 6:3             |                              2197 | 3 1 |   |
    * 6:4             |                             28561 | 3 1 |   |
    * 6:6             |                           4826809 | 3 1 |   |
    * 6:9             |                       10604499373 | 3 1 |   |
    * 6:12            |                    23298085122481 | 3 1 |   |
    * 6:18            |             112455406951957393129 | 3 1 |   |
    * 6:36            |                             13^36 | 3 1 |   |
    *                 |                                   |     |   |
    * 9:3             |                             12167 | 3 1 |   |
    * 9:4             |                            279841 | 3 1 |   |
    * 9:6             |                         148035889 | 3 1 |   |
    * 9:9             |                     1801152661463 | 3 1 |   |
    * 9:12            |                 21914624432020321 | 3 1 |   |
    * 9:18            |         3244150909895248285300369 | 3 1 |   |
    * 9:36            |                             23^36 | 3 1 |   |
    *                 |                                   |     |   |
    * 12:3            |                             50653 | 3 1 |   |
    * 12:4            |                           1874161 | 3 1 |   |
    * 12:6            |                        2565726409 | 3 1 |   |
    * 12:9            |                   129961739795077 | 3 1 |   |
    * 12:12           |               6582952005840035281 | 3 1 |   |
    * 12:18           |     16890053810563300749953435929 | 3 1 |   |
    * 12:36           |                             37^36 | 3 1 |   |
    *                 |                                   |     |   |
    * 18:3            |                            226981 | 3 1 |   |
    * 18:4            |                          13845841 | 3 1 |   |
    * 18:6            |                       51520374361 | 3 1 |   |
    * 18:9            |                 11694146092834141 | 3 1 |   |
    * 18:12           |            2654348974297586158321 | 3 1 |   |
    * 18:18           | 136753052840548005895349735207881 | 3 1 |   |
    * 18:36           |                             61^36 | 3 1 |   |
    *                 |                                   |     |   |
    * 36:3            |                           3442951 | 3 1 |   |
    * 36:4            |                         519885601 | 3 1 |   |
    * 36:6            |                    11853911588401 | 3 1 |   |
    * 36:9            |              40812436757196811351 | 3 1 |   |
    * 36:12           |       140515219945627518837736801 | 3 1 |   |
    * 36:18           |                            151^18 | 3 1 |   |
    * 36:36           |                            151^36 | 3 1 |77 |
    * ----------------+-----------------------------------+-----+---+---
    * The last part is left unsorted to show the method of construction.
    * a (when sorted ) = this sequence
    * h = rote height in gammas = A109301
    * w = rote wayage in gammas = A001221
    * s = count in (h, w) class = A111799
    * t = count in height class = A109300

A111799

Triangle T(h, w) = number of rotes of height h and wayage w.

TeX Array

$\begin{array}{l|*{10}{r}} h \backslash w & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 1 \\ 1 & & 1 \\ 2 & & 3 & 4 \\ 3 & & 77 & ? & ? & ? & ? & ? & ? & ? & ? \end{array}$

ASCII

 Example

    * Table T(h, w), omitting zeros, begins as follows:
    * h\w| 0   1   2   3   4   5   6   7   8   9
    * ---+---------------------------------------
    *  0 | 1
    *  1 |     1
    *  2 |     3   4
    *  3 |    77   ?   ?   ?   ?   ?   ?   ?   ?

A111800

Order of the rote (rooted odd tree with only exponent symmetries) for n.

TeX + JPEG

$\text{Writing}~ \operatorname{prime}(i)^j ~\text{as}~ i\!:\!j, 2500 = 4 \cdot 625 = 2^2 5^4 = 1\!:\!2 ~~ 3\!:\!4 ~\text{has the following rote:}$

$\text{So}~ a(2500) = a(1\!:\!2 ~~ 3\!:\!4) = a(1) + a(2) + a(3) + a(4) + 1 = 1 + 3 + 5 + 5 + 1 = 15.$

ASCII

 Example

    * Writing prime(i)^j as i:j and using equal signs between identified nodes:
    * 2500 = 4 * 625 = 2^2 5^4 = 1:2 3:4 has the following rote:
    *                
    *       o-o   o-o
    *       |     |  
    *   o-o o-o o-o  
    *   |   |   |    
    * o-o   o---o    
    * |     |        
    * O=====O        
    *                
    * So a(2500) = a(1:2 3:4) = a(1)+a(2)+a(3)+a(4)+1 = 1+3+5+5+1 = 15.

A111801

Numbers that have a positive primal code characteristic, that is, positive integers j for which A108352(j) > 0.

A112095

Positive integers sorted by rote weight, rote height, and rote wayage.

TeX Array

$\begin{array}{l|r|rrr|r|r|r} \hline\hline \text{Primal Function} & \text{Primal Code} ~=~ a & g & h & w & r & s & t \\ \hline\hline \varnothing & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ \hline\hline 1\!:\!1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline\hline 2\!:\!1 & 3 & 2 & 2 & 1 & & & \\ 1\!:\!2 & 4 & 2 & 2 & 1 & 2 & 2 & 2 \\ \hline\hline 2\!:\!2 & 9 & 3 & 2 & 1 & 1 & & \\ \hline 1\!:\!1 ~~ 2\!:\!1 & 6 & 3 & 2 & 2 & 1 & 2 & \\ \hline 3\!:\!1 & 5 & 3 & 3 & 1 & & & \\ 4\!:\!1 & 7 & 3 & 3 & 1 & & & \\ 1\!:\!3 & 8 & 3 & 3 & 1 & & & \\ 1\!:\!4 & 16 & 3 & 3 & 1 & 4 & 4 & 6 \\ \hline\hline 1\!:\!2 ~~ 2\!:\!1 & 12 & 4 & 2 & 2 & & & \\ 1\!:\!1 ~~ 2\!:\!2 & 18 & 4 & 2 & 2 & 2 & 2 & \\ \hline 6\!:\!1 & 13 & 4 & 3 & 1 & & & \\ 9\!:\!1 & 23 & 4 & 3 & 1 & & & \\ 3\!:\!2 & 25 & 4 & 3 & 1 & & & \\ 2\!:\!3 & 27 & 4 & 3 & 1 & & & \\ 4\!:\!2 & 49 & 4 & 3 & 1 & & & \\ 1\!:\!6 & 64 & 4 & 3 & 1 & & & \\ 2\!:\!4 & 81 & 4 & 3 & 1 & & & \\ 1\!:\!9 & 512 & 4 & 3 & 1 & 8 & & \\ \hline 1\!:\!1 ~~ 3\!:\!1 & 10 & 4 & 3 & 2 & & & \\ 1\!:\!1 ~~ 4\!:\!1 & 14 & 4 & 3 & 2 & 2 & 10 & \\ \hline 5\!:\!1 & 11 & 4 & 4 & 1 & & & \\ 7\!:\!1 & 17 & 4 & 4 & 1 & & & \\ 8\!:\!1 & 19 & 4 & 4 & 1 & & & \\ 1\!:\!5 & 32 & 4 & 4 & 1 & & & \\ 16\!:\!1 & 53 & 4 & 4 & 1 & & & \\ 1\!:\!7 & 128 & 4 & 4 & 1 & & & \\ 1\!:\!8 & 256 & 4 & 4 & 1 & & & \\ 1\!:\!16 & 65536 & 4 & 4 & 1 & 8 & 8 & 20 \\ \hline\hline 1\!:\!2 ~~ 2\!:\!2 & 36 & 5 & 2 & 2 & 1 & 1 & \\ \hline 12\!:\!1 & 37 & 5 & 3 & 1 & & & \\ 18\!:\!1 & 61 & 5 & 3 & 1 & & & \\ 3\!:\!3 & 125 & 5 & 3 & 1 & & & \\ 6\!:\!2 & 169 & 5 & 3 & 1 & & & \\ 4\!:\!3 & 343 & 5 & 3 & 1 & & & \\ 9\!:\!2 & 529 & 5 & 3 & 1 & & & \\ 3\!:\!4 & 625 & 5 & 3 & 1 & & & \\ 2\!:\!6 & 729 & 5 & 3 & 1 & & & \\ 4\!:\!4 & 2401 & 5 & 3 & 1 & & & \\ 1\!:\!12 & 4096 & 5 & 3 & 1 & & & \\ 2\!:\!9 & 19683 & 5 & 3 & 1 & & & \\ 1\!:\!18 & 262144 & 5 & 3 & 1 & 12 & & \\ \hline 2\!:\!1 ~~ 3\!:\!1 & 15 & 5 & 3 & 2 & & & \\ 1\!:\!2 ~~ 3\!:\!1 & 20 & 5 & 3 & 2 & & & \\ 2\!:\!1 ~~ 4\!:\!1 & 21 & 5 & 3 & 2 & & & \\ 1\!:\!3 ~~ 2\!:\!1 & 24 & 5 & 3 & 2 & & & \\ 1\!:\!1 ~~ 6\!:\!1 & 26 & 5 & 3 & 2 & & & \\ 1\!:\!2 ~~ 4\!:\!1 & 28 & 5 & 3 & 2 & & & \\ 1\!:\!1 ~~ 9\!:\!1 & 46 & 5 & 3 & 2 & & & \\ 1\!:\!4 ~~ 2\!:\!1 & 48 & 5 & 3 & 2 & & & \\ 1\!:\!1 ~~ 3\!:\!2 & 50 & 5 & 3 & 2 & & & \\ 1\!:\!1 ~~ 2\!:\!3 & 54 & 5 & 3 & 2 & & & \\ 1\!:\!1 ~~ 4\!:\!2 & 98 & 5 & 3 & 2 & & & \\ 1\!:\!1 ~~ 2\!:\!4 & 162 & 5 & 3 & 2 & 12 & 24 & \\ \hline 10\!:\!1 & 29 & 5 & 4 & 1 & & & \\ 13\!:\!1 & 41 & 5 & 4 & 1 & & & \\ 14\!:\!1 & 43 & 5 & 4 & 1 & & & \\ 23\!:\!1 & 83 & 5 & 4 & 1 & & & \\ 25\!:\!1 & 97 & 5 & 4 & 1 & & & \\ 27\!:\!1 & 103 & 5 & 4 & 1 & & & \\ 5\!:\!2 & 121 & 5 & 4 & 1 & & & \\ 49\!:\!1 & 227 & 5 & 4 & 1 & & & \\ 2\!:\!5 & 243 & 5 & 4 & 1 & & & \\ 7\!:\!2 & 289 & 5 & 4 & 1 & & & \\ 64\!:\!1 & 311 & 5 & 4 & 1 & & & \\ 8\!:\!2 & 361 & 5 & 4 & 1 & & & \\ 81\!:\!1 & 419 & 5 & 4 & 1 & & & \\ 1\!:\!10 & 1024 & 5 & 4 & 1 & & & \\ 2\!:\!7 & 2187 & 5 & 4 & 1 & & & \\ 16\!:\!2 & 2809 & 5 & 4 & 1 & & & \\ 512\!:\!1 & 3671 & 5 & 4 & 1 & & & \\ 2\!:\!8 & 6561 & 5 & 4 & 1 & & & \\ 1\!:\!13 & 8192 & 5 & 4 & 1 & & & \\ 1\!:\!14 & 16384 & 5 & 4 & 1 & & & \\ 1\!:\!23 & 8388608 & 5 & 4 & 1 & & & \\ 1\!:\!25 & 33554432 & 5 & 4 & 1 & & & \\ 2\!:\!16 & 43046721 & 5 & 4 & 1 & & & \\ 1\!:\!27 & 134217728 & 5 & 4 & 1 & & & \\ 1\!:\!49 & 562949953421312 & 5 & 4 & 1 & & & \\ 1\!:\!64 & 18446744073709551616 & 5 & 4 & 1 & & & \\ 1\!:\!81 & 2417851639229258349412352 & 5 & 4 & 1 & & & \\ 1\!:\!512 & 2^{512} & 5 & 4 & 1 & 28 & & \\ \hline 1\!:\!1 ~~ 5\!:\!1 & 22 & 5 & 4 & 2 & & & \\ 1\!:\!1 ~~ 7\!:\!1 & 34 & 5 & 4 & 2 & & & \\ 1\!:\!1 ~~ 8\!:\!1 & 38 & 5 & 4 & 2 & & & \\ 1\!:\!1 ~~ 16\!:\!1 & 106 & 5 & 4 & 2 & 4 & 32 & \\ \hline 11\!:\!1 & 31 & 5 & 5 & 1 & & & \\ 17\!:\!1 & 59 & 5 & 5 & 1 & & & \\ 19\!:\!1 & 67 & 5 & 5 & 1 & & & \\ 32\!:\!1 & 131 & 5 & 5 & 1 & & & \\ 53\!:\!1 & 241 & 5 & 5 & 1 & & & \\ 128\!:\!1 & 719 & 5 & 5 & 1 & & & \\ 256\!:\!1 & 1619 & 5 & 5 & 1 & & & \\ 1\!:\!11 & 2048 & 5 & 5 & 1 & & & \\ 1\!:\!17 & 131072 & 5 & 5 & 1 & & & \\ 1\!:\!19 & 524288 & 5 & 5 & 1 & & & \\ 65536\!:\!1 & 821641 & 5 & 5 & 1 & & & \\ 1\!:\!32 & 4294967296 & 5 & 5 & 1 & & & \\ 1\!:\!53 & 9007199254740992 & 5 & 5 & 1 & & & \\ 1\!:\!128 & 2^{128} & 5 & 5 & 1 & & & \\ 1\!:\!256 & 2^{256} & 5 & 5 & 1 & & & \\ 1\!:\!65536 & 2^{65536} & 5 & 5 & 1 & 16 & 16 & 73 \\ \hline\hline \end{array}$

ASCII

 Example

    * Table of Primal Functions, Codes, Sort Parameters and Subtotals
    * ================================================================
    * Primal Function |       Primal Code   =   a | g h w | r | s | t
    * ================================================================
    * { }             |                         1 | 0 0 0 | 1 | 1 | 1
    * ================================================================
    * 1:1             |                         2 | 1 1 1 | 1 | 1 | 1
    * ================================================================
    * 2:1             |                         3 | 2 2 1 |   |   |
    * 1:2             |                         4 | 2 2 1 | 2 | 2 | 2
    * ================================================================
    * 2:2             |                         9 | 3 2 1 | 1 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 2:1         |                         6 | 3 2 2 | 1 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 3:1             |                         5 | 3 3 1 |   |   |
    * 4:1             |                         7 | 3 3 1 |   |   |
    * 1:3             |                         8 | 3 3 1 |   |   |
    * 1:4             |                        16 | 3 3 1 | 4 | 4 | 6
    * ================================================================
    * 1:2 2:1         |                        12 | 4 2 2 |   |   |
    * 1:1 2:2         |                        18 | 4 2 2 | 2 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 6:1             |                        13 | 4 3 1 |   |   |
    * 9:1             |                        23 | 4 3 1 |   |   |
    * 3:2             |                        25 | 4 3 1 |   |   |
    * 2:3             |                        27 | 4 3 1 |   |   |
    * 4:2             |                        49 | 4 3 1 |   |   |
    * 1:6             |                        64 | 4 3 1 |   |   |
    * 2:4             |                        81 | 4 3 1 |   |   |
    * 1:9             |                       512 | 4 3 1 | 8 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 3:1         |                        10 | 4 3 2 |   |   |
    * 1:1 4:1         |                        14 | 4 3 2 | 2 |10 |
    * ----------------+---------------------------+-------+---+---+---
    * 5:1             |                        11 | 4 4 1 |   |   |
    * 7:1             |                        17 | 4 4 1 |   |   |
    * 8:1             |                        19 | 4 4 1 |   |   |
    * 1:5             |                        32 | 4 4 1 |   |   |
    * 16:1            |                        53 | 4 4 1 |   |   |
    * 1:7             |                       128 | 4 4 1 |   |   |
    * 1:8             |                       256 | 4 4 1 |   |   |
    * 1:16            |                     65536 | 4 4 1 | 8 | 8 |20
    * ================================================================
    * 1:2 2:2         |                        36 | 5 2 2 | 1 | 1 |
    * ----------------+---------------------------+-------+---+---+---
    * 12:1            |                        37 | 5 3 1 |   |   |
    * 18:1            |                        61 | 5 3 1 |   |   |
    * 3:3             |                       125 | 5 3 1 |   |   |
    * 6:2             |                       169 | 5 3 1 |   |   |
    * 4:3             |                       343 | 5 3 1 |   |   |
    * 9:2             |                       529 | 5 3 1 |   |   |
    * 3:4             |                       625 | 5 3 1 |   |   |
    * 2:6             |                       729 | 5 3 1 |   |   |
    * 4:4             |                      2401 | 5 3 1 |   |   |
    * 1:12            |                      4096 | 5 3 1 |   |   |
    * 2:9             |                     19683 | 5 3 1 |   |   |
    * 1:18            |                    262144 | 5 3 1 |12 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 2:1 3:1         |                        15 | 5 3 2 |   |   |
    * 1:2 3:1         |                        20 | 5 3 2 |   |   |
    * 2:1 4:1         |                        21 | 5 3 2 |   |   |
    * 1:3 2:1         |                        24 | 5 3 2 |   |   |
    * 1:1 6:1         |                        26 | 5 3 2 |   |   |
    * 1:2 4:1         |                        28 | 5 3 2 |   |   |
    * 1:1 9:1         |                        46 | 5 3 2 |   |   |
    * 1:4 2:1         |                        48 | 5 3 2 |   |   |
    * 1:1 3:2         |                        50 | 5 3 2 |   |   |
    * 1:1 2:3         |                        54 | 5 3 2 |   |   |
    * 1:1 4:2         |                        98 | 5 3 2 |   |   |
    * 1:1 2:4         |                       162 | 5 3 2 |12 |24 |
    * ----------------+---------------------------+-------+---+---+---
    * 10:1            |                        29 | 5 4 1 |   |   |
    * 13:1            |                        41 | 5 4 1 |   |   |
    * 14:1            |                        43 | 5 4 1 |   |   |
    * 23:1            |                        83 | 5 4 1 |   |   |
    * 25:1            |                        97 | 5 4 1 |   |   |
    * 27:1            |                       103 | 5 4 1 |   |   |
    * 5:2             |                       121 | 5 4 1 |   |   |
    * 49:1            |                       227 | 5 4 1 |   |   |
    * 2:5             |                       243 | 5 4 1 |   |   |
    * 7:2             |                       289 | 5 4 1 |   |   |
    * 64:1            |                       311 | 5 4 1 |   |   |
    * 8:2             |                       361 | 5 4 1 |   |   |
    * 81:1            |                       419 | 5 4 1 |   |   |
    * 1:10            |                      1024 | 5 4 1 |   |   |
    * 2:7             |                      2187 | 5 4 1 |   |   |
    * 16:2            |                      2809 | 5 4 1 |   |   |
    * 512:1           |                      3671 | 5 4 1 |   |   |
    * 2:8             |                      6561 | 5 4 1 |   |   |
    * 1:13            |                      8192 | 5 4 1 |   |   |
    * 1:14            |                     16384 | 5 4 1 |   |   |
    * 1:23            |                   8388608 | 5 4 1 |   |   |
    * 1:25            |                  33554432 | 5 4 1 |   |   |
    * 2:16            |                  43046721 | 5 4 1 |   |   |
    * 1:27            |                 134217728 | 5 4 1 |   |   |
    * 1:49            |           562949953421312 | 5 4 1 |   |   |
    * 1:64            |      18446744073709551616 | 5 4 1 |   |   |
    * 1:81            | 2417851639229258349412352 | 5 4 1 |   |   |
    * 1:512           |                     2^512 | 5 4 1 |28 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 5:1         |                        22 | 5 4 2 |   |   |
    * 1:1 7:1         |                        34 | 5 4 2 |   |   |
    * 1:1 8:1         |                        38 | 5 4 2 |   |   |
    * 1:1 16:1        |                       106 | 5 4 2 | 4 |32 |
    * ----------------+---------------------------+-------+---+---+---
    * 11:1            |                        31 | 5 5 1 |   |   |
    * 17:1            |                        59 | 5 5 1 |   |   |
    * 19:1            |                        67 | 5 5 1 |   |   |
    * 32:1            |                       131 | 5 5 1 |   |   |
    * 53:1            |                       241 | 5 5 1 |   |   |
    * 128:1           |                       719 | 5 5 1 |   |   |
    * 256:1           |                      1619 | 5 5 1 |   |   |
    * 1:11            |                      2048 | 5 5 1 |   |   |
    * 1:17            |                    131072 | 5 5 1 |   |   |
    * 1:19            |                    524288 | 5 5 1 |   |   |
    * 65536:1         |                    821641 | 5 5 1 |   |   |
    * 1:32            |                4294967296 | 5 5 1 |   |   |
    * 1:53            |          9007199254740992 | 5 5 1 |   |   |
    * 1:128           |                     2^128 | 5 5 1 |   |   |
    * 1:256           |                     2^256 | 5 5 1 |   |   |
    * 1:65536         |                   2^65536 | 5 5 1 |16 |16 |73
    * ================================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * h = rote height in gammas = A109301
    * w = rote wayage in gammas = A001221
    * r = number in (g,h,w) set = A112096
    * s = count in (g, h) class = A111793
    * t = count in weight class = A061396

A112096

Tetrahedron T(g, h, w) = number of rotes of weight g, height h, wayage w.

TeX Array

$\begin{array}{l|*{9}{r}} g \backslash (h,w) & (0,0) & (1,1) & (2,1) & (2,2) & (3,1) & (3,2) & (4,1) & (4,2) & (5,1) \\ \hline 0 & 1~ \\ 1 & & 1~ \\ 2 & & & 2~ \\ 3 & & & 1~ & 1~ & 4~ \\ 4 & & & & 2~ & 8~ & 2~ & 8~ \\ 5 & & & & 1~ & 12~ & 12~ & 28~ & 4~ & 16~ \end{array}$

ASCII

 Example

    * Table T(g, h, w), omitting empty cells, starts out as follows:
    * g\(h,w) | (0,0) (1,1) (2,1) (2,2) (3,1) (3,2) (4,1) (4,2) (5,1)
    * --------+-------------------------------------------------------
    * 0       |   1
    * 1       |         1
    * 2       |               2
    * 3       |               1     1     4
    * 4       |                     2     8     2     8
    * 5       |                     1    12    12    28     4    16

A112480

Positive integers sorted by rote weight, rote wagage, and rote height.

TeX Array

$\begin{array}{l|r|rrr|r|r|r} \hline\hline \text{Primal Function} & \text{Primal Code} ~=~ a & g & w & h & r & s & t \\ \hline\hline \varnothing & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ \hline\hline 1\!:\!1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline\hline 2\!:\!1 & 3 & 2 & 1 & 2 & & & \\ 1\!:\!2 & 4 & 2 & 1 & 2 & 2 & 2 & 2 \\ \hline\hline 2\!:\!2 & 9 & 3 & 1 & 2 & 1 & & \\ \hline 3\!:\!1 & 5 & 3 & 1 & 3 & & & \\ 4\!:\!1 & 7 & 3 & 1 & 3 & & & \\ 1\!:\!3 & 8 & 3 & 1 & 3 & & & \\ 1\!:\!4 &16 & 3 & 1 & 3 & 4 & 5 & \\ \hline 1\!:\!1 ~~ 2\!:\!1 & 6 & 3 & 2 & 2 & 1 & 1 & 6 \\ \hline\hline 6\!:\!1 & 13 & 4 & 1 & 3 & & & \\ 9\!:\!1 & 23 & 4 & 1 & 3 & & & \\ 3\!:\!2 & 25 & 4 & 1 & 3 & & & \\ 2\!:\!3 & 27 & 4 & 1 & 3 & & & \\ 4\!:\!2 & 49 & 4 & 1 & 3 & & & \\ 1\!:\!6 & 64 & 4 & 1 & 3 & & & \\ 2\!:\!4 & 81 & 4 & 1 & 3 & & & \\ 1\!:\!9 & 512 & 4 & 1 & 3 & 8 & & \\ \hline 5\!:\!1 & 11 & 4 & 1 & 4 & & & \\ 7\!:\!1 & 17 & 4 & 1 & 4 & & & \\ 8\!:\!1 & 19 & 4 & 1 & 4 & & & \\ 1\!:\!5 & 32 & 4 & 1 & 4 & & & \\ 16\!:\!1 & 53 & 4 & 1 & 4 & & & \\ 1\!:\!7 & 128 & 4 & 1 & 4 & & & \\ 1\!:\!8 & 256 & 4 & 1 & 4 & & & \\ 1\!:\!16 & 65536 & 4 & 1 & 4 & 8 & 16 & \\ \hline 1\!:\!2 2\!:\!1 & 12 & 4 & 2 & 2 & & & \\ 1\!:\!1 2\!:\!2 & 18 & 4 & 2 & 2 & 2 & & \\ \hline 1\!:\!1 3\!:\!1 & 10 & 4 & 2 & 3 & & & \\ 1\!:\!1 4\!:\!1 & 14 & 4 & 2 & 3 & 2 & 4 & 20 \\ \hline\hline 12\!:\!1 & 37 & 5 & 1 & 3 & & & \\ 18\!:\!1 & 61 & 5 & 1 & 3 & & & \\ 3\!:\!3 & 125 & 5 & 1 & 3 & & & \\ 6\!:\!2 & 169 & 5 & 1 & 3 & & & \\ 4\!:\!3 & 343 & 5 & 1 & 3 & & & \\ 9\!:\!2 & 529 & 5 & 1 & 3 & & & \\ 3\!:\!4 & 625 & 5 & 1 & 3 & & & \\ 2\!:\!6 & 729 & 5 & 1 & 3 & & & \\ 4\!:\!4 & 2401 & 5 & 1 & 3 & & & \\ 1\!:\!12 & 4096 & 5 & 1 & 3 & & & \\ 2\!:\!9 & 19683 & 5 & 1 & 3 & & & \\ 1\!:\!18 & 262144 & 5 & 1 & 3 &12 & & \\ \hline 10\!:\!1 & 29 & 5 & 1 & 4 & & & \\ 13\!:\!1 & 41 & 5 & 1 & 4 & & & \\ 14\!:\!1 & 43 & 5 & 1 & 4 & & & \\ 23\!:\!1 & 83 & 5 & 1 & 4 & & & \\ 25\!:\!1 & 97 & 5 & 1 & 4 & & & \\ 27\!:\!1 & 103 & 5 & 1 & 4 & & & \\ 5\!:\!2 & 121 & 5 & 1 & 4 & & & \\ 49\!:\!1 & 227 & 5 & 1 & 4 & & & \\ 2\!:\!5 & 243 & 5 & 1 & 4 & & & \\ 7\!:\!2 & 289 & 5 & 1 & 4 & & & \\ 64\!:\!1 & 311 & 5 & 1 & 4 & & & \\ 8\!:\!2 & 361 & 5 & 1 & 4 & & & \\ 81\!:\!1 & 419 & 5 & 1 & 4 & & & \\ 1\!:\!10 & 1024 & 5 & 1 & 4 & & & \\ 2\!:\!7 & 2187 & 5 & 1 & 4 & & & \\ 16\!:\!2 & 2809 & 5 & 1 & 4 & & & \\ 512\!:\!1 & 3671 & 5 & 1 & 4 & & & \\ 2\!:\!8 & 6561 & 5 & 1 & 4 & & & \\ 1\!:\!13 & 8192 & 5 & 1 & 4 & & & \\ 1\!:\!14 & 16384 & 5 & 1 & 4 & & & \\ 1\!:\!23 & 8388608 & 5 & 1 & 4 & & & \\ 1\!:\!25 & 33554432 & 5 & 1 & 4 & & & \\ 2\!:\!16 & 43046721 & 5 & 1 & 4 & & & \\ 1\!:\!27 & 134217728 & 5 & 1 & 4 & & & \\ 1\!:\!49 & 562949953421312 & 5 & 1 & 4 & & & \\ 1\!:\!64 & 18446744073709551616 & 5 & 1 & 4 & & & \\ 1\!:\!81 & 2417851639229258349412352 & 5 & 1 & 4 & & & \\ 1\!:\!512 & 2^{512} & 5 & 1 & 4 & 28 & & \\ \hline 11\!:\!1 & 31 & 5 & 1 & 5 & & & \\ 17\!:\!1 & 59 & 5 & 1 & 5 & & & \\ 19\!:\!1 & 67 & 5 & 1 & 5 & & & \\ 32\!:\!1 & 131 & 5 & 1 & 5 & & & \\ 53\!:\!1 & 241 & 5 & 1 & 5 & & & \\ 128\!:\!1 & 719 & 5 & 1 & 5 & & & \\ 256\!:\!1 & 1619 & 5 & 1 & 5 & & & \\ 1\!:\!11 & 2048 & 5 & 1 & 5 & & & \\ 1\!:\!17 & 131072 & 5 & 1 & 5 & & & \\ 1\!:\!19 & 524288 & 5 & 1 & 5 & & & \\ 65536\!:\!1 & 821641 & 5 & 1 & 5 & & & \\ 1\!:\!32 & 4294967296 & 5 & 1 & 5 & & & \\ 1\!:\!53 & 9007199254740992 & 5 & 1 & 5 & & & \\ 1\!:\!128 & 2^{128} & 5 & 1 & 5 & & & \\ 1\!:\!256 & 2^{256} & 5 & 1 & 5 & & & \\ 1\!:\!65536 & 2^{65536} & 5 & 1 & 5 & 16 & 56 & \\ \hline 1\!:\!2 ~~ 2\!:\!2 & 36 & 5 & 2 & 2 & 1 & & \\ \hline 2\!:\!1 ~~ 3\!:\!1 & 15 & 5 & 2 & 3 & & & \\ 1\!:\!2 ~~ 3\!:\!1 & 20 & 5 & 2 & 3 & & & \\ 2\!:\!1 ~~ 4\!:\!1 & 21 & 5 & 2 & 3 & & & \\ 1\!:\!3 ~~ 2\!:\!1 & 24 & 5 & 2 & 3 & & & \\ 1\!:\!1 ~~ 6\!:\!1 & 26 & 5 & 2 & 3 & & & \\ 1\!:\!2 ~~ 4\!:\!1 & 28 & 5 & 2 & 3 & & & \\ 1\!:\!1 ~~ 9\!:\!1 & 46 & 5 & 2 & 3 & & & \\ 1\!:\!4 ~~ 2\!:\!1 & 48 & 5 & 2 & 3 & & & \\ 1\!:\!1 ~~ 3\!:\!2 & 50 & 5 & 2 & 3 & & & \\ 1\!:\!1 ~~ 2\!:\!3 & 54 & 5 & 2 & 3 & & & \\ 1\!:\!1 ~~ 4\!:\!2 & 98 & 5 & 2 & 3 & & & \\ 1\!:\!1 ~~ 2\!:\!4 & 162 & 5 & 2 & 3 & 12 & & \\ \hline 1\!:\!1 ~~ 5\!:\!1 & 22 & 5 & 2 & 4 & & & \\ 1\!:\!1 ~~ 7\!:\!1 & 34 & 5 & 2 & 4 & & & \\ 1\!:\!1 ~~ 8\!:\!1 & 38 & 5 & 2 & 4 & & & \\ 1\!:\!1 ~~ 16\!:\!1 & 106 & 5 & 2 & 4 & 4 & 17 & 73 \\ \hline\hline \end{array}$

ASCII

 Example

    * Table of Primal Functions, Codes, Sort Parameters and Subtotals
    * ================================================================
    * Primal Function |       Primal Code   =   a | g w h | r | s | t
    * ================================================================
    * { }             |                         1 | 0 0 0 | 1 | 1 | 1
    * ================================================================
    * 1:1             |                         2 | 1 1 1 | 1 | 1 | 1
    * ================================================================
    * 2:1             |                         3 | 2 1 2 |   |   |
    * 1:2             |                         4 | 2 1 2 | 2 | 2 | 2
    * ================================================================
    * 2:2             |                         9 | 3 1 2 | 1 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 3:1             |                         5 | 3 1 3 |   |   |
    * 4:1             |                         7 | 3 1 3 |   |   |
    * 1:3             |                         8 | 3 1 3 |   |   |
    * 1:4             |                        16 | 3 1 3 | 4 | 5 |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 2:1         |                         6 | 3 2 2 | 1 | 1 | 6
    * ================================================================
    * 6:1             |                        13 | 4 1 3 |   |   |
    * 9:1             |                        23 | 4 1 3 |   |   |
    * 3:2             |                        25 | 4 1 3 |   |   |
    * 2:3             |                        27 | 4 1 3 |   |   |
    * 4:2             |                        49 | 4 1 3 |   |   |
    * 1:6             |                        64 | 4 1 3 |   |   |
    * 2:4             |                        81 | 4 1 3 |   |   |
    * 1:9             |                       512 | 4 1 3 | 8 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 5:1             |                        11 | 4 1 4 |   |   |
    * 7:1             |                        17 | 4 1 4 |   |   |
    * 8:1             |                        19 | 4 1 4 |   |   |
    * 1:5             |                        32 | 4 1 4 |   |   |
    * 16:1            |                        53 | 4 1 4 |   |   |
    * 1:7             |                       128 | 4 1 4 |   |   |
    * 1:8             |                       256 | 4 1 4 |   |   |
    * 1:16            |                     65536 | 4 1 4 | 8 |16 |
    * ----------------+---------------------------+-------+---+---+---
    * 1:2 2:1         |                        12 | 4 2 2 |   |   |
    * 1:1 2:2         |                        18 | 4 2 2 | 2 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 3:1         |                        10 | 4 2 3 |   |   |
    * 1:1 4:1         |                        14 | 4 2 3 | 2 | 4 |20
    * ================================================================
    * 12:1            |                        37 | 5 1 3 |   |   |
    * 18:1            |                        61 | 5 1 3 |   |   |
    * 3:3             |                       125 | 5 1 3 |   |   |
    * 6:2             |                       169 | 5 1 3 |   |   |
    * 4:3             |                       343 | 5 1 3 |   |   |
    * 9:2             |                       529 | 5 1 3 |   |   |
    * 3:4             |                       625 | 5 1 3 |   |   |
    * 2:6             |                       729 | 5 1 3 |   |   |
    * 4:4             |                      2401 | 5 1 3 |   |   |
    * 1:12            |                      4096 | 5 1 3 |   |   |
    * 2:9             |                     19683 | 5 1 3 |   |   |
    * 1:18            |                    262144 | 5 1 3 |12 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 10:1            |                        29 | 5 1 4 |   |   |
    * 13:1            |                        41 | 5 1 4 |   |   |
    * 14:1            |                        43 | 5 1 4 |   |   |
    * 23:1            |                        83 | 5 1 4 |   |   |
    * 25:1            |                        97 | 5 1 4 |   |   |
    * 27:1            |                       103 | 5 1 4 |   |   |
    * 5:2             |                       121 | 5 1 4 |   |   |
    * 49:1            |                       227 | 5 1 4 |   |   |
    * 2:5             |                       243 | 5 1 4 |   |   |
    * 7:2             |                       289 | 5 1 4 |   |   |
    * 64:1            |                       311 | 5 1 4 |   |   |
    * 8:2             |                       361 | 5 1 4 |   |   |
    * 81:1            |                       419 | 5 1 4 |   |   |
    * 1:10            |                      1024 | 5 1 4 |   |   |
    * 2:7             |                      2187 | 5 1 4 |   |   |
    * 16:2            |                      2809 | 5 1 4 |   |   |
    * 512:1           |                      3671 | 5 1 4 |   |   |
    * 2:8             |                      6561 | 5 1 4 |   |   |
    * 1:13            |                      8192 | 5 1 4 |   |   |
    * 1:14            |                     16384 | 5 1 4 |   |   |
    * 1:23            |                   8388608 | 5 1 4 |   |   |
    * 1:25            |                  33554432 | 5 1 4 |   |   |
    * 2:16            |                  43046721 | 5 1 4 |   |   |
    * 1:27            |                 134217728 | 5 1 4 |   |   |
    * 1:49            |           562949953421312 | 5 1 4 |   |   |
    * 1:64            |      18446744073709551616 | 5 1 4 |   |   |
    * 1:81            | 2417851639229258349412352 | 5 1 4 |   |   |
    * 1:512           |                     2^512 | 5 1 4 |28 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 11:1            |                        31 | 5 1 5 |   |   |
    * 17:1            |                        59 | 5 1 5 |   |   |
    * 19:1            |                        67 | 5 1 5 |   |   |
    * 32:1            |                       131 | 5 1 5 |   |   |
    * 53:1            |                       241 | 5 1 5 |   |   |
    * 128:1           |                       719 | 5 1 5 |   |   |
    * 256:1           |                      1619 | 5 1 5 |   |   |
    * 1:11            |                      2048 | 5 1 5 |   |   |
    * 1:17            |                    131072 | 5 1 5 |   |   |
    * 1:19            |                    524288 | 5 1 5 |   |   |
    * 65536:1         |                    821641 | 5 1 5 |   |   |
    * 1:32            |                4294967296 | 5 1 5 |   |   |
    * 1:53            |          9007199254740992 | 5 1 5 |   |   |
    * 1:128           |                     2^128 | 5 1 5 |   |   |
    * 1:256           |                     2^256 | 5 1 5 |   |   |
    * 1:65536         |                   2^65536 | 5 1 5 |16 |56 |
    * ----------------+---------------------------+-------+---+---+---
    * 1:2 2:2         |                        36 | 5 2 2 | 1 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 2:1 3:1         |                        15 | 5 2 3 |   |   |
    * 1:2 3:1         |                        20 | 5 2 3 |   |   |
    * 2:1 4:1         |                        21 | 5 2 3 |   |   |
    * 1:3 2:1         |                        24 | 5 2 3 |   |   |
    * 1:1 6:1         |                        26 | 5 2 3 |   |   |
    * 1:2 4:1         |                        28 | 5 2 3 |   |   |
    * 1:1 9:1         |                        46 | 5 2 3 |   |   |
    * 1:4 2:1         |                        48 | 5 2 3 |   |   |
    * 1:1 3:2         |                        50 | 5 2 3 |   |   |
    * 1:1 2:3         |                        54 | 5 2 3 |   |   |
    * 1:1 4:2         |                        98 | 5 2 3 |   |   |
    * 1:1 2:4         |                       162 | 5 2 3 |12 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 5:1         |                        22 | 5 2 4 |   |   |
    * 1:1 7:1         |                        34 | 5 2 4 |   |   |
    * 1:1 8:1         |                        38 | 5 2 4 |   |   |
    * 1:1 16:1        |                       106 | 5 2 4 | 4 |17 |73
    * ================================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * w = rote wayage in gammas = A001221
    * h = rote height in gammas = A109301
    * r = number in (g,h,w) set = A112481
    * s = count in (g, w) class = A111797
    * t = count in weight class = A061396

A112481

Tetrahedron T(g, w, h) = number of rotes of weight g, wayage w, height h.

TeX Array

$\begin{array}{l|*{9}{r}} g \backslash (w,h) & (0,0) & (1,1) & (1,2) & & (1,3) & & (1,4) & & (1,5) \\ & & & & (2,2) & & (2,3) & & (2,4) & \\ \hline\hline 0 & 1~ \\ \hline 1 & & 1~ \\ \hline 2 & & & 2~ \\ \hline 3 & & & 1~ & & 4~ \\ & & & & 1~ & \\ \hline 4 & & & & & 8~ & & 8~ \\ & & & & 2~ & & 2~ & \\ \hline 5 & & & & & 12~ & & 28~ & & 16~ \\ & & & & 1~ & & 12~ & & 4~ & \\ \hline \end{array}$

ASCII

 Example

    * Table T(g, w, h), omitting empty cells, starts out as follows:
    * --------+-------------------------------------------------------
    * g\(w,h) | (0,0) (1,1) (1,2)       (1,3)       (1,4)       (1,5)
    *         |                   (2,2)       (2,3)       (2,4)      
    * ========+=======================================================
    * 0       |   1                                                  
    * --------+-------------------------------------------------------
    * 1       |         1                                            
    * --------+-------------------------------------------------------
    * 2       |               2                                      
    * --------+-------------------------------------------------------
    * 3       |               1           4                          
    * 3       |                     1                                
    * --------+-------------------------------------------------------
    * 4       |                           8           8              
    * 4       |                     2           2                    
    * --------+-------------------------------------------------------
    * 5       |                          12          28          16  
    * 5       |                     1          12           4        
    * --------+-------------------------------------------------------
    * Row sums = A111797. Horizontal section sums = A061396.

A112846

Number of riffs on n or fewer nodes. Number of rotes on 2n+1 or fewer nodes.

A112868

Positive integers sorted by rote weight and primal code characteristic.

TeX Array

$\begin{array}{l|r|rr|r|r} \hline\hline \text{Primal Function} & \text{Primal Code} ~=~ a & g & q & s & t \\ \hline\hline \varnothing & 1 & 0 & 1 & 1 & 1 \\ \hline\hline 1\!:\!1 & 2 & 1 & 0 & 1 & 1 \\ \hline\hline 2\!:\!1 & 3 & 2 & 2 & & \\ 1\!:\!2 & 4 & 2 & 2 & 2 & 2 \\ \hline\hline 1\!:\!1 ~~ 2\!:\!1 & 6 & 3 & 0 & & \\ 2\!:\!2 & 9 & 3 & 0 & 2 & \\ \hline 3\!:\!1 & 5 & 3 & 2 & & \\ 4\!:\!1 & 7 & 3 & 2 & & \\ 1\!:\!3 & 8 & 3 & 2 & & \\ 1\!:\!4 & 16 & 3 & 2 & 4 & 6 \\ \hline\hline 1\!:\!1 ~~ 3\!:\!1 & 10 & 4 & 0 & & \\ 1\!:\!2 ~~ 2\!:\!1 & 12 & 4 & 0 & & \\ 1\!:\!1 ~~ 4\!:\!1 & 14 & 4 & 0 & & \\ 1\!:\!1 ~~ 2\!:\!2 & 18 & 4 & 0 & 4 & \\ \hline 5\!:\!1 & 11 & 4 & 2 & & \\ 6\!:\!1 & 13 & 4 & 2 & & \\ 7\!:\!1 & 17 & 4 & 2 & & \\ 8\!:\!1 & 19 & 4 & 2 & & \\ 9\!:\!1 & 23 & 4 & 2 & & \\ 3\!:\!2 & 25 & 4 & 2 & & \\ 2\!:\!3 & 27 & 4 & 2 & & \\ 1\!:\!5 & 32 & 4 & 2 & & \\ 4\!:\!2 & 49 & 4 & 2 & & \\ 16\!:\!1 & 53 & 4 & 2 & & \\ 1\!:\!6 & 64 & 4 & 2 & & \\ 2\!:\!4 & 81 & 4 & 2 & & \\ 1\!:\!7 & 128 & 4 & 2 & & \\ 1\!:\!8 & 256 & 4 & 2 & & \\ 1\!:\!9 & 512 & 4 & 2 & & \\ 1\!:\!16 & 65536 & 4 & 2 & 16 & 20 \\ \hline\hline \end{array}$

ASCII

 Example

    * Primal Functions, Primal Codes, Sort Parameters, Subtotals
    * ==========================================================
    * Primal Function |       Primal Code   =   a | g q | s | t
    * ==========================================================
    * { }             |                         1 | 0 1 | 1 | 1
    * ==========================================================
    * 1:1             |                         2 | 1 0 | 1 | 1
    * ==========================================================
    * 2:1             |                         3 | 2 2 |   |
    * 1:2             |                         4 | 2 2 | 2 | 2
    * ==========================================================
    * 1:1 2:1         |                         6 | 3 0 |   |
    * 2:2             |                         9 | 3 0 | 2 |
    * ----------------+---------------------------+-----+---+---
    * 3:1             |                         5 | 3 2 |   |
    * 4:1             |                         7 | 3 2 |   |
    * 1:3             |                         8 | 3 2 |   |
    * 1:4             |                        16 | 3 2 | 4 | 6
    * ==========================================================
    * 1:1 3:1         |                        10 | 4 0 |   |
    * 1:2 2:1         |                        12 | 4 0 |   |
    * 1:1 4:1         |                        14 | 4 0 |   |
    * 1:1 2:2         |                        18 | 4 0 | 4 |
    * ----------------+---------------------------+-----+---+---
    * 5:1             |                        11 | 4 2 |   |
    * 6:1             |                        13 | 4 2 |   |
    * 7:1             |                        17 | 4 2 |   |
    * 8:1             |                        19 | 4 2 |   |
    * 9:1             |                        23 | 4 2 |   |
    * 3:2             |                        25 | 4 2 |   |
    * 2:3             |                        27 | 4 2 |   |
    * 1:5             |                        32 | 4 2 |   |
    * 4:2             |                        49 | 4 2 |   |
    * 16:1            |                        53 | 4 2 |   |
    * 1:6             |                        64 | 4 2 |   |
    * 2:4             |                        81 | 4 2 |   |
    * 1:7             |                       128 | 4 2 |   |
    * 1:8             |                       256 | 4 2 |   |
    * 1:9             |                       512 | 4 2 |   |
    * 1:16            |                     65536 | 4 2 |16 |20
    * ==========================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * q = primal code character = A108352
    * s = count in (g, q) class = A112869
    * t = count in weight class = A061396

A112869

Triangle T(g, q) = number of rotes of weight g and primal code characteristic q.

TeX Array

$\begin{array}{l|rrrrrr} g \backslash q & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & & 1 \\ 1 & 1 & \\ 2 & & & 2 \\ 3 & 2 & & 4 \\ 4 & 4 & & 16 \\ 5 & 13 & & 56 & 4 \end{array}$

ASCII

 Example

    * Table T(g, q), omitting empty cells, begins as follows:
    * g\q| 0   1   2   3   4   5
    * ---+-----------------------
    *  0 |     1
    *  1 | 1
    *  2 |         2
    *  3 | 2       4
    *  4 | 4      16
    *  5 |13      56   4

A112870

Positive integers sorted by rote height and primal code characteristic.

TeX Array

$\begin{array}{l|r|rr|r|r} \text{Primal Function} & \text{Primal Code} ~=~ a & h & q & s & t \\ \hline \varnothing & 1 & 0 & 1 & 1 & 1 \\ \hline 1\!:\!1 & 2 & 1 & 0 & 1 & 1 \\ \hline 1\!:\!1 ~~ 2\!:\!1 & 6 & 2 & 0 & & \\ 2\!:\!2 & 9 & 2 & 0 & & \\ 1\!:\!2 ~~ 2\!:\!1 & 12 & 2 & 0 & & \\ 1\!:\!1 ~~ 2\!:\!2 & 18 & 2 & 0 & & \\ 1\!:\!2 ~~ 2\!:\!2 & 36 & 2 & 0 & 5 & \\ \hline 2\!:\!1 & 3 & 2 & 2 & & \\ 1\!:\!2 & 4 & 2 & 2 & 2 & 7 \\ \hline \end{array}$

ASCII

 Example

    * Primal Function | Primal Code = a | h q | s | t
    * ----------------+-----------------+-----+---+---
    * { }             |               1 | 0 1 | 1 | 1
    * ----------------+-----------------+-----+---+---
    * 1:1             |               2 | 1 0 | 1 | 1
    * ----------------+-----------------+-----+---+---
    * 1:1 2:1         |               6 | 2 0 |   |
    * 2:2             |               9 | 2 0 |   |
    * 1:2 2:1         |              12 | 2 0 |   |
    * 1:1 2:2         |              18 | 2 0 |   |
    * 1:2 2:2         |              36 | 2 0 | 5 |
    * ----------------+-----------------+-----+---+---
    * 2:1             |               3 | 2 2 |   |
    * 1:2             |               4 | 2 2 | 2 | 7
    * ----------------+-----------------+-----+---+---
    * a = this sequence
    * h = rote height in gammas = A109301
    * q = primal code character = A108352
    * s = count in (h, q) class = A112871
    * t = count in height class = A109300

A112871

Triangle T(h, q) = number of rotes of height h and quench q.

TeX Array

$\begin{array}{l|rrr} h \backslash q & 0 & 1 & 2 \\ \hline 0 & & 1 & \\ 1 & 1 & & \\ 2 & 5 & & 2 \\ \end{array}$

ASCII

 Example

    * Table T(h, q), omitting empty cells, begins as follows:
    * h\q| 0   1   2
    * ---+----------
    *  0 |     1    
    *  1 | 1        
    *  2 | 5       2
    * Row sums = A109300.

A112872

First differences of A061396.

A113197

Positive integers sorted by rote weight, rote height, and rote quench.

TeX Array

$\begin{array}{l|r|rrr|r|r|r} \hline\hline \text{Primal Function} & \text{Primal Code} ~=~ a & g & h & q & r & s & t \\ \hline\hline \varnothing & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\ \hline\hline 1\!:\!1 & 2 & 1 & 1 & 0 & 1 & 1 & 1 \\ \hline\hline 2\!:\!1 & 3 & 2 & 2 & 2 & & & \\ 1\!:\!2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\ \hline\hline 1\!:\!1 ~~ 2\!:\!1 & 6 & 3 & 2 & 0 & & & \\ 2\!:\!2 & 9 & 3 & 2 & 0 & 2 & 2 & \\ \hline 3\!:\!1 & 5 & 3 & 3 & 2 & & & \\ 4\!:\!1 & 7 & 3 & 3 & 2 & & & \\ 1\!:\!3 & 8 & 3 & 3 & 2 & & & \\ 1\!:\!4 & 16 & 3 & 3 & 2 & 4 & 4 & 6 \\ \hline\hline 1\!:\!2 ~~ 2\!:\!1 & 12 & 4 & 2 & 0 & & & \\ 1\!:\!1 ~~ 2\!:\!2 & 18 & 4 & 2 & 0 & 2 & 2 & \\ \hline 1\!:\!1 ~~ 3\!:\!1 & 10 & 4 & 3 & 0 & & & \\ 1\!:\!1 ~~ 4\!:\!1 & 14 & 4 & 3 & 0 & 2 & & \\ \hline 6\!:\!1 & 13 & 4 & 3 & 2 & & & \\ 9\!:\!1 & 23 & 4 & 3 & 2 & & & \\ 3\!:\!2 & 25 & 4 & 3 & 2 & & & \\ 2\!:\!3 & 27 & 4 & 3 & 2 & & & \\ 4\!:\!2 & 49 & 4 & 3 & 2 & & & \\ 1\!:\!6 & 64 & 4 & 3 & 2 & & & \\ 2\!:\!4 & 81 & 4 & 3 & 2 & & & \\ 1\!:\!9 & 512 & 4 & 3 & 2 & 8 & 10 & \\ \hline 5\!:\!1 & 11 & 4 & 4 & 2 & & & \\ 7\!:\!1 & 17 & 4 & 4 & 2 & & & \\ 8\!:\!1 & 19 & 4 & 4 & 2 & & & \\ 1\!:\!5 & 32 & 4 & 4 & 2 & & & \\ 16\!:\!1 & 53 & 4 & 4 & 2 & & & \\ 1\!:\!7 & 128 & 4 & 4 & 2 & & & \\ 1\!:\!8 & 256 & 4 & 4 & 2 & & & \\ 1\!:\!16 & 65536 & 4 & 4 & 2 & 8 & 8 & 20 \\ \hline\hline \end{array}$

ASCII

 Example

    * Primal Functions, Primal Codes, Sort Parameters and Subtotals
    * ================================================================
    * Primal Function |       Primal Code   =   a | g h q | r | s | t
    * ================================================================
    * { }             |                         1 | 0 0 1 | 1 | 1 | 1
    * ================================================================
    * 1:1             |                         2 | 1 1 0 | 1 | 1 | 1
    * ================================================================
    * 2:1             |                         3 | 2 2 2 |   |   |
    * 1:2             |                         4 | 2 2 2 | 2 | 2 | 2
    * ================================================================
    * 1:1 2:1         |                         6 | 3 2 0 |   |   |
    * 2:2             |                         9 | 3 2 0 | 2 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 3:1             |                         5 | 3 3 2 |   |   |
    * 4:1             |                         7 | 3 3 2 |   |   |
    * 1:3             |                         8 | 3 3 2 |   |   |
    * 1:4             |                        16 | 3 3 2 | 4 | 4 | 6
    * ================================================================
    * 1:2 2:1         |                        12 | 4 2 0 |   |   |
    * 1:1 2:2         |                        18 | 4 2 0 | 2 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 3:1         |                        10 | 4 3 0 |   |   |
    * 1:1 4:1         |                        14 | 4 3 0 | 2 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 6:1             |                        13 | 4 3 2 |   |   |
    * 9:1             |                        23 | 4 3 2 |   |   |
    * 3:2             |                        25 | 4 3 2 |   |   |
    * 2:3             |                        27 | 4 3 2 |   |   |
    * 4:2             |                        49 | 4 3 2 |   |   |
    * 1:6             |                        64 | 4 3 2 |   |   |
    * 2:4             |                        81 | 4 3 2 |   |   |
    * 1:9             |                       512 | 4 3 2 | 8 |10 |
    * ----------------+---------------------------+-------+---+---+---
    * 5:1             |                        11 | 4 4 2 |   |   |
    * 7:1             |                        17 | 4 4 2 |   |   |
    * 8:1             |                        19 | 4 4 2 |   |   |
    * 1:5             |                        32 | 4 4 2 |   |   |
    * 16:1            |                        53 | 4 4 2 |   |   |
    * 1:7             |                       128 | 4 4 2 |   |   |
    * 1:8             |                       256 | 4 4 2 |   |   |
    * 1:16            |                     65536 | 4 4 2 | 8 | 8 |20
    * ================================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * h = rote height in gammas = A109301
    * q = primal code character = A108352
    * r = number in (g,h,q) set = A113198
    * s = count in (g, h) class = A111793
    * t = count in weight class = A061396

A113198

Tetrahedron T(g, h, q) = number of rotes of weight g, height h, quench q.

TeX Array

$\begin{array}{l|*{10}{r}} \hline \\ g \backslash (h,q) & (0,1) \\ & & (0,1) \\ & & & (2,0) & (2,2) \\ & & & & & (3,0) & (3,2) & (3,3) \\ & & & & & & & & (4,0) & (4,2) \\ & & & & & & & & & & (5,2) \\ \hline\hline 0 & 1~ \\ \hline 1 & & 1~ \\ \hline 2 & & & & 2~ \\ \hline 3 & & & 2~ & & & \\ & & & & & & 4~ \\ \hline 4 & & & 2~ & & & & & & \\ & & & & & 2~ & 8~ & & & \\ & & & & & & & & & 8~ \\ \hline 5 & & & 1~ & & & & & & & \\ & & & & & 8~ & 12~ & 4~ & & & \\ & & & & & & & & 4~ & 28~ & \\ & & & & & & & & & & 16~ \\ \hline \end{array}$

ASCII

 Example

    * Table T(g, h, q), omitting empty cells, starts out as follows:
    * --------+------------------------------------------------------------
    * g\(h,q) | (0,1)                                                      
    *         |       (1,0)                                                
    *         |             (2,0) (2,2)                                    
    *         |                         (3,0) (3,2) (3,3)                  
    *         |                                           (4,0) (4,2)      
    *         |                                                       (5,2)
    * ========+============================================================
    * 0       |   1                                                        
    * --------+------------------------------------------------------------
    * 1       |         1                                                  
    * --------+------------------------------------------------------------
    * 2       |                     2                                      
    * --------+------------------------------------------------------------
    * 3       |               2                                            
    * 3       |                                 4                          
    * --------+------------------------------------------------------------
    * 4       |               2                                            
    * 4       |                           2     8                          
    * 4       |                                                   8        
    * --------+------------------------------------------------------------
    * 5       |               1                                            
    * 5       |                           8    12     4                    
    * 5       |                                             4    28        
    * 5       |                                                        16  
    * --------+------------------------------------------------------------
    * Row sums = A111793. Horizontal section sums = A061396.

A113199

Positive integers sorted by rote weight, rote quench, and rote height.

TeX Array

$\begin{array}{l|r|rrr|r|r|r} \hline\hline \text{Primal Function} & \text{Primal Code} ~=~ a & g & q & h & r & s & t \\ \hline\hline \varnothing & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ \hline\hline 1\!:\!1 & 2 & 1 & 0 & 1 & 1 & 1 & 1 \\ \hline\hline 2\!:\!1 & 3 & 2 & 2 & 2 & & & \\ 1\!:\!2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\ \hline\hline 1\!:\!1 ~~ 2\!:\!1 & 6 & 3 & 0 & 2 & & & \\ 2\!:\!2 & 9 & 3 & 0 & 2 & 2 & 2 & \\ \hline 3\!:\!1 & 5 & 3 & 2 & 3 & & & \\ 4\!:\!1 & 7 & 3 & 2 & 3 & & & \\ 1\!:\!3 & 8 & 3 & 2 & 3 & & & \\ 1\!:\!4 & 16 & 3 & 2 & 3 & 4 & 4 & 6 \\ \hline\hline 1\!:\!2 ~~ 2\!:\!1 & 12 & 4 & 0 & 2 & & & \\ 1\!:\!1 ~~ 2\!:\!2 & 18 & 4 & 0 & 2 & 2 & & \\ \hline 1\!:\!1 ~~ 3\!:\!1 & 10 & 4 & 0 & 3 & & & \\ 1\!:\!1 ~~ 4\!:\!1 & 14 & 4 & 0 & 3 & 2 & 4 & \\ \hline 6\!:\!1 & 13 & 4 & 2 & 3 & & & \\ 9\!:\!1 & 23 & 4 & 2 & 3 & & & \\ 3\!:\!2 & 25 & 4 & 2 & 3 & & & \\ 2\!:\!3 & 27 & 4 & 2 & 3 & & & \\ 4\!:\!2 & 49 & 4 & 2 & 3 & & & \\ 1\!:\!6 & 64 & 4 & 2 & 3 & & & \\ 2\!:\!4 & 81 & 4 & 2 & 3 & & & \\ 1\!:\!9 & 512 & 4 & 2 & 3 & 8 & & \\ \hline 5\!:\!1 & 11 & 4 & 2 & 4 & & & \\ 7\!:\!1 & 17 & 4 & 2 & 4 & & & \\ 8\!:\!1 & 19 & 4 & 2 & 4 & & & \\ 1\!:\!5 & 32 & 4 & 2 & 4 & & & \\ 16\!:\!1 & 53 & 4 & 2 & 4 & & & \\ 1\!:\!7 & 128 & 4 & 2 & 4 & & & \\ 1\!:\!8 & 256 & 4 & 2 & 4 & & & \\ 1\!:\!16 & 65536 & 4 & 2 & 4 & 8 & 16 & 20 \\ \hline\hline \end{array}$

ASCII

 Example

    * Primal Functions, Primal Codes, Sort Parameters and Subtotals
    * ================================================================
    * Primal Function |       Primal Code   =   a | g q h | r | s | t
    * ================================================================
    * { }             |                         1 | 0 1 0 | 1 | 1 | 1
    * ================================================================
    * 1:1             |                         2 | 1 0 1 | 1 | 1 | 1
    * ================================================================
    * 2:1             |                         3 | 2 2 2 |   |   |
    * 1:2             |                         4 | 2 2 2 | 2 | 2 | 2
    * ================================================================
    * 1:1 2:1         |                         6 | 3 0 2 |   |   |
    * 2:2             |                         9 | 3 0 2 | 2 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 3:1             |                         5 | 3 2 3 |   |   |
    * 4:1             |                         7 | 3 2 3 |   |   |
    * 1:3             |                         8 | 3 2 3 |   |   |
    * 1:4             |                        16 | 3 2 3 | 4 | 4 | 6
    * ================================================================
    * 1:2 2:1         |                        12 | 4 0 2 |   |   |
    * 1:1 2:2         |                        18 | 4 0 2 | 2 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 3:1         |                        10 | 4 0 3 |   |   |
    * 1:1 4:1         |                        14 | 4 0 3 | 2 | 4 |
    * ----------------+---------------------------+-------+---+---+---
    * 6:1             |                        13 | 4 2 3 |   |   |
    * 9:1             |                        23 | 4 2 3 |   |   |
    * 3:2             |                        25 | 4 2 3 |   |   |
    * 2:3             |                        27 | 4 2 3 |   |   |
    * 4:2             |                        49 | 4 2 3 |   |   |
    * 1:6             |                        64 | 4 2 3 |   |   |
    * 2:4             |                        81 | 4 2 3 |   |   |
    * 1:9             |                       512 | 4 2 3 | 8 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 5:1             |                        11 | 4 2 4 |   |   |
    * 7:1             |                        17 | 4 2 4 |   |   |
    * 8:1             |                        19 | 4 2 4 |   |   |
    * 1:5             |                        32 | 4 2 4 |   |   |
    * 16:1            |                        53 | 4 2 4 |   |   |
    * 1:7             |                       128 | 4 2 4 |   |   |
    * 1:8             |                       256 | 4 2 4 |   |   |
    * 1:16            |                     65536 | 4 2 4 | 8 |16 |20
    * ================================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * q = primal code character = A108352
    * h = rote height in gammas = A109301
    * r = number in (g,q,h) set = A113200
    * s = count in (g, q) class = A112869
    * t = count in weight class = A061396

A113200

Tetrahedron T(g, q, h) = number of rotes of weight g, quench q, height h.

TeX Array

$\begin{array}{l|*{10}{r}} \hline \\ g \backslash (q,h) & (1,0) & (0,1) & (0,2) & & (0,3) & & & (0,4) & & \\ & & & & (2,2) & & (2,3) & & & (2,4) & (2,5) \\ & & & & & & & (3,3) & & & \\ \hline\hline 0 & 1~ \\ \hline 1 & & 1~ \\ \hline 2 & & & & 2~ \\ \hline 3 & & & 2~ & & \\ & & & & & & 4~ \\ \hline 4 & & & 2~ & & 2~ & & \\ & & & & & & 8~ & & & 8~ \\ \hline 5 & & & 1~ & & 8~ & & & 4~ & 28~ & 16~ \\ & & & & & & 12~ & & & & \\ & & & & & & & 4~ & & & \\ \hline \end{array}$

ASCII

 Example

    * Table T(g, q, h), omitting empty cells, starts out as follows:
    * --------+------------------------------------------------------------
    * g\(q,h) | (1,0) (0,1) (0,2)       (0,3)             (0,4)            
    *         |                   (2,2)       (2,3)             (2,4) (2,5)
    *         |                                     (3,3)                  
    * ========+============================================================
    * 0       |   1                                                        
    * --------+------------------------------------------------------------
    * 1       |         1                                                  
    * --------+------------------------------------------------------------
    * 2       |                     2                                      
    * --------+------------------------------------------------------------
    * 3       |               2                                            
    * 3       |                                 4                          
    * --------+------------------------------------------------------------
    * 4       |               2           2                                
    * 4       |                                 8                 8        
    * --------+------------------------------------------------------------
    * 5       |               1           8                 4              
    * 5       |                                12                28    16  
    * 5       |                                       4                    
    * --------+------------------------------------------------------------
    * Row sums = A112869. Horizontal section sums = A061396.

									1		1
								2		1		2
							3		1		1		3
						4		1		2		1		4
					5		1		3		1		1		5
				6		1		1		1		4		1		6
			7		1		5		2		9		1		1		7
		8		1		6		1		1		1		2		1		8
	9		1		7		1		25		1		3		1		1		9
10		1		1		1		36		1		2		1		8		1		10

															1		1
														2		·		2
													3		·		·		3
												4		·		2		·		4
											5		·		3		1		·		5
										6		·		1		1		4		·		6
									7		·		5		2		9		1		·		7
								8		·		6		1		1		1		2		·		8
							9		·		7		1		25		1		3		1		·		9
						10		·		1		1		36		1		2		1		8		·		10
					11		·		1		1		49		1		5		1		27		1		·		11
				12		·		10		3		1		1		6		1		1		1		2		·		12
			13		·		11		1		1		2		7		1		125		4		3		1		·		13
		14		·		3		1		100		1		1		1		216		1		1		1		4		·		14
	15		·		13		2		121		1		3		1		343		1		5		1		9		1		·		15
16		·		14		1		9		1		10		1		1		1		6		1		2		1		2		·		16

															1		1
														2		1		2
													3		2		1		3
												4		3		2		1		4
											5		4		1		2		1		5
										6		5		1		1		2		1		6
									7		6		1		1		1		2		1		7
								8		7		6		1		1		1		2		1		8
							9		8		1		6		1		1		1		2		1		9
						10		9		1		1		6		1		1		1		2		1		10
					11		10		9		1		1		6		1		1		1		2		1		11
				12		11		10		9		1		1		6		1		1		1		2		1		12
			13		12		1		10		9		1		1		6		1		1		1		2		1		13
		14		13		18		1		10		9		1		1		6		1		1		1		2		1		14
	15		14		1		12		1		10		9		1		1		6		1		1		1		2		1		15
16		15		14		1		18		1		10		9		1		1		6		1		1		1		2		1		16

									1		1
								2		1		2
							3		1		1		3
						4		1		2		1		4
					5		1		3		1		1		5
				6		1		1		1		4		1		6
			7		1		5		2		9		1		1		7
		8		1		6		1		1		1		2		1		8
	9		1		7		1		25		1		3		1		1		9
10		1		1		1		36		1		2		1		8		1		10

															1		1
														2		·		2
													3		·		·		3
												4		·		2		·		4
											5		·		3		1		·		5
										6		·		1		1		4		·		6
									7		·		5		2		9		1		·		7
								8		·		6		1		1		1		2		·		8
							9		·		7		1		25		1		3		1		·		9
						10		·		1		1		36		1		2		1		8		·		10
					11		·		1		1		49		1		5		1		27		1		·		11
				12		·		10		3		1		1		6		1		1		1		2		·		12
			13		·		11		1		1		2		7		1		125		4		3		1		·		13
		14		·		3		1		100		1		1		1		216		1		1		1		4		·		14
	15		·		13		2		121		1		3		1		343		1		5		1		9		1		·		15
16		·		14		1		9		1		10		1		1		1		6		1		2		1		2		·		16

															1		1
														2		1		2
													3		2		1		3
												4		3		2		1		4
											5		4		1		2		1		5
										6		5		1		1		2		1		6
									7		6		1		1		1		2		1		7
								8		7		6		1		1		1		2		1		8
							9		8		1		6		1		1		1		2		1		9
						10		9		1		1		6		1		1		1		2		1		10
					11		10		9		1		1		6		1		1		1		2		1		11
				12		11		10		9		1		1		6		1		1		1		2		1		12
			13		12		1		10		9		1		1		6		1		1		1		2		1		13
		14		13		18		1		10		9		1		1		6		1		1		1		2		1		14
	15		14		1		12		1		10		9		1		1		6		1		1		1		2		1		15
16		15		14		1		18		1		10		9		1		1		6		1		1		1		2		1		16

User:Jon Awbrey/WORKSPACE

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									1		1
								2		1		2
							3		1		1		3
						4		1		2		1		4
					5		1		3		1		1		5
				6		1		1		1		4		1		6
			7		1		5		2		9		1		1		7
		8		1		6		1		1		1		2		1		8
	9		1		7		1		25		1		3		1		1		9
10		1		1		1		36		1		2		1		8		1		10

															1		1
														2		·		2
													3		·		·		3
												4		·		2		·		4
											5		·		3		1		·		5
										6		·		1		1		4		·		6
									7		·		5		2		9		1		·		7
								8		·		6		1		1		1		2		·		8
							9		·		7		1		25		1		3		1		·		9
						10		·		1		1		36		1		2		1		8		·		10
					11		·		1		1		49		1		5		1		27		1		·		11
				12		·		10		3		1		1		6		1		1		1		2		·		12
			13		·		11		1		1		2		7		1		125		4		3		1		·		13
		14		·		3		1		100		1		1		1		216		1		1		1		4		·		14
	15		·		13		2		121		1		3		1		343		1		5		1		9		1		·		15
16		·		14		1		9		1		10		1		1		1		6		1		2		1		2		·		16

															1		1
														2		1		2
													3		2		1		3
												4		3		2		1		4
											5		4		1		2		1		5
										6		5		1		1		2		1		6
									7		6		1		1		1		2		1		7
								8		7		6		1		1		1		2		1		8
							9		8		1		6		1		1		1		2		1		9
						10		9		1		1		6		1		1		1		2		1		10
					11		10		9		1		1		6		1		1		1		2		1		11
				12		11		10		9		1		1		6		1		1		1		2		1		12
			13		12		1		10		9		1		1		6		1		1		1		2		1		13
		14		13		18		1		10		9		1		1		6		1		1		1		2		1		14
	15		14		1		12		1		10		9		1		1		6		1		1		1		2		1		15
16		15		14		1		18		1		10		9		1		1		6		1		1		1		2		1		16