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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)

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Contents

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

    Rote 1 Big.jpg  
Riff 2 Big.jpg Rote 2 Big.jpg

Riff 3 Big.jpg Rote 3 Big.jpg

Riff 4 Big.jpg Rote 4 Big.jpg

Riff 5 Big.jpg Rote 5 Big.jpg

Riff 6 Big.jpg Rote 6 Big.jpg

Riff 7 Big.jpg Rote 7 Big.jpg

Riff 8 Big.jpg Rote 8 Big.jpg

Riff 9 Big.jpg Rote 9 Big.jpg

Riff 16 Big.jpg Rote 16 Big.jpg

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

JPEG

    Rote 1 Big.jpg
Riff 2 Big.jpg Rote 2 Big.jpg

Riff 3 Big.jpg Rote 3 Big.jpg

Riff 4 Big.jpg Rote 4 Big.jpg

Riff 5 Big.jpg Rote 5 Big.jpg

Riff 6 Big.jpg Rote 6 Big.jpg

Riff 7 Big.jpg Rote 7 Big.jpg

Riff 8 Big.jpg Rote 8 Big.jpg

Riff 9 Big.jpg Rote 9 Big.jpg

Riff 16 Big.jpg Rote 16 Big.jpg

Riff 10 Big.jpg Rote 10 Big.jpg

Riff 11 Big.jpg Rote 11 Big.jpg

Riff 12 Big.jpg Rote 12 Big.jpg

Riff 13 Big.jpg Rote 13 Big.jpg

Riff 14 Big.jpg Rote 14 Big.jpg

Riff 17 Big.jpg Rote 17 Big.jpg

Riff 18 Big.jpg Rote 18 Big.jpg

Riff 19 Big.jpg Rote 19 Big.jpg

Riff 23 Big.jpg Rote 23 Big.jpg

Riff 25 Big.jpg Rote 25 Big.jpg

Riff 27 Big.jpg Rote 27 Big.jpg

Riff 32 Big.jpg Rote 32 Big.jpg

Riff 49 Big.jpg Rote 49 Big.jpg

Riff 53 Big.jpg Rote 53 Big.jpg

Riff 64 Big.jpg Rote 64 Big.jpg

Riff 81 Big.jpg Rote 81 Big.jpg

Riff 128 Big.jpg Rote 128 Big.jpg

Riff 256 Big.jpg Rote 256 Big.jpg

Riff 512 Big.jpg Rote 512 Big.jpg

Riff 65536 Big.jpg Rote 65536 Big.jpg

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

 



Riff 2 Big.jpg



Riff 3 Big.jpg



Riff 4 Big.jpg



Riff 5 Big.jpg



Riff 6 Big.jpg



Riff 7 Big.jpg



Riff 8 Big.jpg



Riff 9 Big.jpg



Riff 10 Big.jpg



Riff 11 Big.jpg



Riff 12 Big.jpg



Riff 13 Big.jpg



Riff 14 Big.jpg



Riff 15 Big.jpg



Riff 16 Big.jpg



Riff 17 Big.jpg



Riff 18 Big.jpg



Riff 19 Big.jpg



Riff 20 Big.jpg



Riff 21 Big.jpg



Riff 22 Big.jpg



Riff 23 Big.jpg



Riff 24 Big.jpg



Riff 25 Big.jpg



Riff 26 Big.jpg



Riff 27 Big.jpg



Riff 28 Big.jpg



Riff 29 Big.jpg



Riff 30 Big.jpg



Riff 31 Big.jpg



Riff 32 Big.jpg



Riff 33 Big.jpg



Riff 34 Big.jpg



Riff 35 Big.jpg



Riff 36 Big.jpg



Riff 37 Big.jpg



Riff 38 Big.jpg



Riff 39 Big.jpg



Riff 40 Big.jpg



Riff 41 Big.jpg



Riff 42 Big.jpg



Riff 43 Big.jpg



Riff 44 Big.jpg



Riff 45 Big.jpg



Riff 46 Big.jpg



Riff 47 Big.jpg



Riff 48 Big.jpg



Riff 49 Big.jpg



Riff 50 Big.jpg



Riff 51 Big.jpg



Riff 52 Big.jpg



Riff 53 Big.jpg



Riff 54 Big.jpg



Riff 55 Big.jpg



Riff 56 Big.jpg



Riff 57 Big.jpg



Riff 58 Big.jpg



Riff 59 Big.jpg



Riff 60 Big.jpg



A062860

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

Wiki + TeX + JPEG

 



Riff 2 Big.jpg



Riff 3 Big.jpg



Riff 5 Big.jpg



Riff 10 Big.jpg



Riff 15 Big.jpg



Riff 30 Big.jpg



Riff 55 Big.jpg



Riff 105 Big.jpg



Riff 165 Big.jpg



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

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

Array

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

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

TeX 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

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

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

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

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

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.

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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

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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

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