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# User:Jon Awbrey/WORKSPACE

## Articles

• Graph Theory, Arithmetic
• Graph Theory, Computation
• Graph Theory, Logic
• Differential Logic
• Functional Logic
• Information Theory

## Work Notes

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

## 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

${\displaystyle {\text{Prime Factorizations, Riffs, Rotes, and Traversals}}\!}$
 ${\displaystyle {\text{Integer}}\!}$ ${\displaystyle {\text{Factorization}}\!}$ ${\displaystyle {\text{Notation}}\!}$ ${\displaystyle {\text{Riff Digraph}}\!}$ ${\displaystyle {\text{Rote Graph}}\!}$ ${\displaystyle {\text{Traversal}}\!}$
 ${\displaystyle 1\!}$ ${\displaystyle 1\!}$
 ${\displaystyle 2\!}$ ${\displaystyle {\text{p}}_{1}^{1}\!}$ ${\displaystyle {\text{p}}\!}$ ${\displaystyle ((~))}$
 ${\displaystyle 3\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{2}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{\text{p}}\!}$ ${\displaystyle (((~))(~))}$ ${\displaystyle 4\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{2}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}^{\text{p}}\!}$ ${\displaystyle ((((~))))}$
 ${\displaystyle 5\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{3}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[10pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}_{\text{p}}}\!}$ ${\displaystyle ((((~))(~))(~))}$ ${\displaystyle 6\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}{\text{p}}_{\text{p}}\!}$ ${\displaystyle ((~))(((~))(~))}$ ${\displaystyle 7\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{4}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[10pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}^{\text{p}}}\!}$ ${\displaystyle (((((~))))(~))}$ ${\displaystyle 8\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{3}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{1}}\\[10pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}_{\text{p}}}\!}$ ${\displaystyle (((((~))(~))))}$ ${\displaystyle 9\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{2}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}_{\text{p}}^{\text{p}}\!}$ ${\displaystyle (((~))(((~))))}$ ${\displaystyle 16\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{4}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{2}}\\[10pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}^{\text{p}}}\!}$ ${\displaystyle ((((((~))))))}$

### 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--------------------------------------------------------------------------------

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

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

### JPEG

${\displaystyle {\text{Prime Factorizations, Riffs, and Rotes}}\!}$
 ${\displaystyle {\text{Integer}}\!}$ ${\displaystyle {\text{Factorization}}\!}$ ${\displaystyle {\text{Notation}}\!}$ ${\displaystyle {\text{Riff Digraph}}\!}$ ${\displaystyle {\text{Rote Graph}}\!}$
 ${\displaystyle 1\!}$ ${\displaystyle 1\!}$
 ${\displaystyle 2\!}$ ${\displaystyle {\text{p}}_{1}^{1}\!}$ ${\displaystyle {\text{p}}\!}$
 ${\displaystyle 3\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{2}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{\text{p}}\!}$ ${\displaystyle 4\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{2}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}^{\text{p}}\!}$
 ${\displaystyle 5\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{3}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}_{\text{p}}}\!}$ ${\displaystyle 6\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}{\text{p}}_{\text{p}}\!}$ ${\displaystyle 7\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{4}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}^{\text{p}}}\!}$ ${\displaystyle 8\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{3}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}_{\text{p}}}\!}$ ${\displaystyle 9\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{2}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}_{\text{p}}^{\text{p}}\!}$ ${\displaystyle 16\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{4}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{2}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}^{\text{p}}}\!}$
 ${\displaystyle 10\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{3}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[12pt]&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!}$ ${\displaystyle 11\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{5}^{1}&=&{\text{p}}_{{\text{p}}_{3}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{2}^{1}}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!}$ ${\displaystyle 12\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{2}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}^{\text{p}}{\text{p}}_{\text{p}}\!}$ ${\displaystyle 13\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{6}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!}$ ${\displaystyle 14\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{4}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[12pt]&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!}$ ${\displaystyle 17\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{7}^{1}&=&{\text{p}}_{{\text{p}}_{4}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{2}}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}\!}$ ${\displaystyle 18\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{2}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}{\text{p}}_{\text{p}}^{\text{p}}\!}$ ${\displaystyle 19\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{8}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{3}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{2}^{1}}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}^{{\text{p}}_{\text{p}}}}\!}$ ${\displaystyle 23\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{9}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{2}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}_{\text{p}}^{\text{p}}}\!}$ ${\displaystyle 25\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{3}^{2}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{{\text{p}}_{1}^{1}}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}_{\text{p}}}^{\text{p}}\!}$ ${\displaystyle 27\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{2}^{3}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{2}^{1}}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}_{\text{p}}^{{\text{p}}_{\text{p}}}\!}$ ${\displaystyle 32\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{5}&=&{\text{p}}_{1}^{{\text{p}}_{3}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{2}^{1}}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}_{{\text{p}}_{\text{p}}}}\!}$ ${\displaystyle 49\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{4}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{{\text{p}}_{1}^{1}}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{{\text{p}}_{1}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}^{\text{p}}}^{\text{p}}\!}$ ${\displaystyle 53\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{16}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{4}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{2}}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}}^{1}\end{array}}}$ ${\displaystyle {\text{p}}_{{\text{p}}^{{\text{p}}^{\text{p}}}}\!}$ ${\displaystyle 64\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{6}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}{\text{p}}_{\text{p}}}\!}$ ${\displaystyle 81\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{2}^{4}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{2}}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}}$ ${\displaystyle {\text{p}}_{\text{p}}^{{\text{p}}^{\text{p}}}\!}$ ${\displaystyle 128\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{7}&=&{\text{p}}_{1}^{{\text{p}}_{4}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{2}}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}_{{\text{p}}^{\text{p}}}}\!}$ ${\displaystyle 256\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{8}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{3}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{2}^{1}}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}^{{\text{p}}_{\text{p}}}}\!}$ ${\displaystyle 512\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{9}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{2}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}_{\text{p}}^{\text{p}}}\!}$ ${\displaystyle 65536\!}$ ${\displaystyle {\begin{array}{lll}{\text{p}}_{1}^{16}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{4}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{2}}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}}\end{array}}}$ ${\displaystyle {\text{p}}^{{\text{p}}^{{\text{p}}^{\text{p}}}}\!}$

### 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

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

## A062860

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

### Wiki + TeX + JPEG

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

## 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

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

### 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

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

#### Array

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

#### Matrix

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

### 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

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

### 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.

### TeX Array

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

### 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

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

### 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

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

### 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

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

### 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

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

### 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

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

### 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

 ${\displaystyle {\begin{array}{l}2\!:\!1\\3\end{array}}}$ ${\displaystyle {\begin{array}{l}1\!:\!2\\4\end{array}}}$ ${\displaystyle {\begin{array}{l}1\!:\!1~~2\!:\!1\\6\end{array}}}$ ${\displaystyle {\begin{array}{l}2\!:\!2\\9\end{array}}}$ ${\displaystyle {\begin{array}{l}1\!:\!2~~2\!:\!1\\12\end{array}}}$ ${\displaystyle {\begin{array}{l}1\!:\!1~~2\!:\!2\\18\end{array}}}$ ${\displaystyle {\begin{array}{l}1\!:\!2~~2\!:\!2\\36\end{array}}}$

### 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

${\displaystyle 802701=9\cdot 89189={\text{p}}_{2}^{2}{\text{p}}_{8638}^{1}}$
${\displaystyle {\text{Writing}}~(\operatorname {prime} (i))^{j}~{\text{as}}~i\!:\!j,~{\text{we have:}}}$
${\displaystyle {\begin{array}{lllll}802701&=&9\cdot 89189&=&2\!:\!2~~8638\!:\!1\\8638&=&2\cdot 7\cdot 617&=&1\!:\!1~~4\!:\!1~~113\!:\!1\\113&&&=&30\!:\!1\\30&=&2\cdot 3\cdot 5&=&1\!:\!1~~2\!:\!1~~3\!:\!1\\4&&&=&1\!:\!2\\3&&&=&2\!:\!1\\2&&&=&1\!:\!1\end{array}}}$
${\displaystyle {\text{So the rote of 802701 is the following graph:}}\!}$
${\displaystyle {\text{By inspection, the rote height of 802701 is 6.}}\!}$

### JPEG

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

### 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

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

### Wiki Table

 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 hth row = A007097. Number of values in the hth row = A109300(h). Number of values up through the hth 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

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