# User:Jon Awbrey/SEQUENCES

## A061396

### Plain Wiki Table

#### Large Scale

 $$\text{Integer}\!$$ $$\text{Factorization}\!$$ $$\text{Notation}\!$$ $$\text{Riff Digraph}\!$$ $$\text{Rote Graph}\!$$ $$\text{Traversal}\!$$ $$1\!$$ $$1\!$$ $$2\!$$ $$\text{p}_1^1\!$$ $$\text{p}\!$$ $$((~))$$ $$3\!$$ $$\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}$$ $$\text{p}_\text{p}\!$$ $$(((~))(~))$$ $$4\!$$ $$\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}$$ $$\text{p}^\text{p}\!$$ $$((((~))))$$ $$5\!$$ \begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \'"UNIQ-MathJax1-QINU"' '"UNIQ-MathJax2-QINU"' '"UNIQ-MathJax3-QINU"' '"UNIQ-MathJax4-QINU"' :{| border="1" cellpadding="20" | [[Image:Rote 802701 Big.jpg|330px]] |} '"UNIQ-MathJax5-QINU"' ==='"UNIQ--h-40--QINU"'JPEG=== {| align="center" border="1" cellpadding="6" | valign="bottom" | [[Image:Rote 1 Big.jpg|20px]] \(1\! $$a(1) ~=~ 0$$ $$\text{p}\!$$ $$a(2) ~=~ 1$$ $$\text{p}_\text{p}\!$$ $$a(3) ~=~ 2$$ $$\text{p}^\text{p}\!$$ $$a(4) ~=~ 2$$ $$\text{p}_{\text{p}_\text{p}}\!$$ $$a(5) ~=~ 3$$ $$\text{p} \text{p}_\text{p}\!$$ $$a(6) ~=~ 2$$ $$\text{p}_{\text{p}^\text{p}}\!$$ $$a(7) ~=~ 3$$ $$\text{p}^{\text{p}_\text{p}}\!$$ $$a(8) ~=~ 3$$ $$\text{p}_\text{p}^\text{p}\!$$ $$a(9) ~=~ 2$$ $$\text{p} \text{p}_{\text{p}_\text{p}}\!$$ $$a(10) ~=~ 3$$ $$\text{p}_{\text{p}_{\text{p}_\text{p}}}\!$$ $$a(11) ~=~ 4$$ $$\text{p}^\text{p} \text{p}_\text{p}\!$$ $$a(12) ~=~ 2$$ $$\text{p}_{\text{p} \text{p}_\text{p}}\!$$ $$a(13) ~=~ 3$$ $$\text{p} \text{p}_{\text{p}^\text{p}}\!$$ $$a(14) ~=~ 3$$ $$\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$$ $$a(15) ~=~ 3$$ $$\text{p}^{\text{p}^\text{p}}\!$$ $$a(16) ~=~ 3$$ $$\text{p}_{\text{p}_{\text{p}^\text{p}}}\!$$ $$a(17) ~=~ 4$$ $$\text{p} \text{p}_\text{p}^\text{p}\!$$ $$a(18) ~=~ 2$$ $$\text{p}_{\text{p}^{\text{p}_\text{p}}}\!$$ $$a(19) ~=~ 4$$ $$\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$$ $$a(20) ~=~ 3$$ $$\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$$ $$a(21) ~=~ 3$$ $$\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$$ $$a(22) ~=~ 4$$ $$\text{p}_{\text{p}_\text{p}^\text{p}}\!$$ $$a(23) ~=~ 3$$ $$\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!$$ $$a(24) ~=~ 3$$ $$\text{p}_{\text{p}_\text{p}}^\text{p}\!$$ $$a(25) ~=~ 3$$ $$\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$$ $$a(26) ~=~ 3$$ $$\text{p}_\text{p}^{\text{p}_\text{p}}\!$$ $$a(27) ~=~ 3$$ $$\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!$$ $$a(28) ~=~ 3$$ $$\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$$ $$a(29) ~=~ 4$$ $$\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$$ $$a(30) ~=~ 3$$ $$\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!$$ $$a(31) ~=~ 5$$ $$\text{p}^{\text{p}_{\text{p}_\text{p}}}\!$$ $$a(32) ~=~ 4$$ $$\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$$ $$a(33) ~=~ 4$$ $$\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$$ $$a(34) ~=~ 4$$ $$\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$$ $$a(35) ~=~ 3$$ $$\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!$$ $$a(36) ~=~ 2$$ $$\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!$$ $$a(37) ~=~ 3$$ $$\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$$ $$a(38) ~=~ 4$$ $$\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$$ $$a(39) ~=~ 3$$ $$\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!$$ $$a(40) ~=~ 3$$ $$\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!$$ $$a(41) ~=~ 4$$ $$\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!$$ $$a(42) ~=~ 3$$ $$\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!$$ $$a(43) ~=~ 4$$ $$\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$$ $$a(44) ~=~ 4$$ $$\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!$$ $$a(45) ~=~ 3$$ $$\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!$$ $$a(46) ~=~ 3$$ $$\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!$$ $$a(47) ~=~ 4$$ $$\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!$$ $$a(48) ~=~ 3$$ $$\text{p}_{\text{p}^\text{p}}^\text{p}\!$$ $$a(49) ~=~ 3$$ $$\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!$$ $$a(50) ~=~ 3$$ $$\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!$$ $$a(51) ~=~ 4$$ $$\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!$$ $$a(52) ~=~ 3$$ $$\text{p}_{\text{p}^{\text{p}^\text{p}}}\!$$ $$a(53) ~=~ 4$$ $$\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!$$ $$a(54) ~=~ 3$$ $$\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!$$ $$a(55) ~=~ 4$$ $$\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!$$ $$a(56) ~=~ 3$$ $$\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!$$ $$a(57) ~=~ 4$$ $$\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!$$ $$a(58) ~=~ 4$$ $$\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!$$ $$a(59) ~=~ 5$$ $$\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!$$ $$a(60) ~=~ 3$$

### ASCII

 Comment

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

Example

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


## A111795

### JPEG

 $$\begin{array}{l} \varnothing \\ 1 \end{array}$$ $$\begin{array}{l} 1\!:\!1 \\ 2 \end{array}$$ $$\begin{array}{l} 2\!:\!1 \\ 3 \end{array}$$ $$\begin{array}{l} 1\!:\!2 \\ 4 \end{array}$$ $$\begin{array}{l} 3\!:\!1 \\ 5 \end{array}$$ $$\begin{array}{l} 4\!:\!1 \\ 7 \end{array}$$ $$\begin{array}{l} 1\!:\!3 \\ 8 \end{array}$$ $$\begin{array}{l} 5\!:\!1 \\ 11 \end{array}$$ $$\begin{array}{l} 1\!:\!4 \\ 16 \end{array}$$ $$\begin{array}{l} 7\!:\!1 \\ 17 \end{array}$$ $$\begin{array}{l} 8\!:\!1 \\ 19 \end{array}$$ $$\begin{array}{l} 11\!:\!1 \\ 31 \end{array}$$ $$\begin{array}{l} 1\!:\!5 \\ 32 \end{array}$$ $$\begin{array}{l} 16\!:\!1 \\ 53 \end{array}$$ $$\begin{array}{l} 17\!:\!1 \\ 59 \end{array}$$

### ASCII

 Example

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


## A111800

### TeX + JPEG

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

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

### ASCII

 Example

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