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Latest revision as of 18:48, 31 January 2010
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" |
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
<p>\(1\!\) \(a(1) ~=~ 0\) |
\(\text{p}\!\) \(a(2) ~=~ 1\) |
\(\text{p}_\text{p}\!\) \(a(3) ~=~ 2\) |
\(\text{p}^\text{p}\!\) \(a(4) ~=~ 2\) |
\(\text{p}_{\text{p}_\text{p}}\!\) \(a(5) ~=~ 3\) |
|
\(\text{p} \text{p}_\text{p}\!\) \(a(6) ~=~ 2\) |
\(\text{p}_{\text{p}^\text{p}}\!\) \(a(7) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}}\!\) \(a(8) ~=~ 3\) |
\(\text{p}_\text{p}^\text{p}\!\) \(a(9) ~=~ 2\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(10) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(11) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(a(12) ~=~ 2\) |
\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(13) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(14) ~=~ 3\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(15) ~=~ 3\) | |
|
\(\text{p}^{\text{p}^\text{p}}\!\) \(a(16) ~=~ 3\) |
\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(17) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(18) ~=~ 2\) |
\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(19) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(20) ~=~ 3\) | |
|
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(21) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(22) ~=~ 4\) |
\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(23) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(a(24) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(25) ~=~ 3\) | |
|
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(26) ~=~ 3\) |
\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(27) ~=~ 3\) |
\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(28) ~=~ 3\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(29) ~=~ 4\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(30) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(a(31) ~=~ 5\) |
\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(a(32) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(33) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(34) ~=~ 4\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(35) ~=~ 3\) | |
|
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(36) ~=~ 2\) |
\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(a(37) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(38) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(39) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(a(40) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(a(41) ~=~ 4\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(42) ~=~ 3\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(a(43) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(44) ~=~ 4\) |
\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(45) ~=~ 3\) | |
|
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(46) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(47) ~=~ 4\) |
\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(a(48) ~=~ 3\) |
\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(a(49) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(50) ~=~ 3\) | |
|
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(51) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(52) ~=~ 3\) |
\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(a(53) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(54) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(55) ~=~ 4\) | |
|
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(56) ~=~ 3\) |
\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(57) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(58) ~=~ 4\) |
\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(a(59) ~=~ 5\) |
\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(60) ~=~ 3\) |
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.
