Difference between revisions of "User:Jon Awbrey/SEQUENCES"
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===Example=== | ===Example=== | ||
| − | <p><math>802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1</math></p> | + | : <p><math>802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1</math></p> |
| − | <p><math>\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}</math></p> | + | : <p><math>\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}</math></p> |
| − | <p><math>\begin{array}{lllll} | + | : <p><math>\begin{array}{lllll} |
802701 | 802701 | ||
& = & 9 \cdot 89189 | & = & 9 \cdot 89189 | ||
| Line 2,541: | Line 2,541: | ||
\end{array}</math></p> | \end{array}</math></p> | ||
| − | <p><math>\text{So | + | : <p><math>\text{So the rote of 802701 is the following graph:}</math></p> |
| − | {| | + | :{| border="1" cellpadding="20" |
| [[Image:Rote 802701 Big.jpg|330px]] | | [[Image:Rote 802701 Big.jpg|330px]] | ||
|} | |} | ||
| − | + | : <p><math>\text{By inspection, the rote height of 802701 is 6.}</math></p> | |
| − | <p><math>\text{ | ||
===JPEG=== | ===JPEG=== | ||
Revision as of 15:48, 26 January 2010
A061396
Plain Wiki Table
Large Scale
| \(\text{Integer}\!\) | \(\text{Factorization}\!\) | \(\text{Notation}\!\) | \(\text{Riff Digraph}\!\) | \(\text{Rote Graph}\!\) | \(\text{Traversal}\!\) |
| \(1\!\) | \(1\!\) | 
|
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| \(2\!\) | \(\text{p}_1^1\!\) | \(\text{p}\!\) | 
|
|
\(((~))\) |
| \(3\!\) |
\(\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}\) |
\(\text{p}_\text{p}\!\) | 
|
|
\((((~))(~))\) |
| \(4\!\) |
\(\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}\) |
\(\text{p}^\text{p}\!\) | 
|
|
\(((((~))))\) |
| \(5\!\) |
\(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[6pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\) |
\(\text{p}_{\text{p}_{\text{p}}}\!\) | 
|
|
\(((((~))(~))(~))\) |
| \(6\!\) |
\(\begin{array}{lll} \text{p}_1^1 \text{p}_2^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 \end{array}\) |
\(\text{p} \text{p}_{\text{p}}\!\) | 
|
|
\(((~))(((~))(~))\) |
| \(7\!\) |
\(\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[6pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\) |
\(\text{p}_{\text{p}^{\text{p}}}\!\) | 
|
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\((((((~))))(~))\) |
| \(8\!\) |
\(\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[6pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}\) |
\(\text{p}^{\text{p}_{\text{p}}}\!\) | 
|
|
\((((((~))(~))))\) |
| \(9\!\) |
\(\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\) |
\(\text{p}_\text{p}^\text{p}\!\) | 
|
|
\((((~))(((~))))\) |
| \(16\!\) |
\(\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[6pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}\) |
\(\text{p}^{\text{p}^{\text{p}}}\!\) | 
|
|
\(((((((~))))))\) |
Small Scale
| \(\text{Integer}\!\) | \(\text{Factorization}\!\) | \(\text{Notation}\!\) | \(\text{Riff Digraph}\!\) | \(\text{Rote Graph}\!\) | \(\text{Traversal}\!\) |
| \(1\!\) | \(1\!\) |
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| \(2\!\) | \(\text{p}_1^1\!\) | \(\text{p}\!\) |
|
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\(((~))\) |
| \(3\!\) |
\(\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}\) |
\(\text{p}_\text{p}\!\) |
|
|
\((((~))(~))\) |
| \(4\!\) |
\(\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}\) |
\(\text{p}^\text{p}\!\) |
|
|
\(((((~))))\) |
| \(5\!\) |
\(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[6pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\) |
\(\text{p}_{\text{p}_{\text{p}}}\!\) |
|
|
\(((((~))(~))(~))\) |
| \(6\!\) |
\(\begin{array}{lll} \text{p}_1^1 \text{p}_2^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 \end{array}\) |
\(\text{p} \text{p}_{\text{p}}\!\) |
|
|
\(((~))(((~))(~))\) |
| \(7\!\) |
\(\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[6pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\) |
\(\text{p}_{\text{p}^{\text{p}}}\!\) |
|
|
\((((((~))))(~))\) |
| \(8\!\) |
\(\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[6pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}\) |
\(\text{p}^{\text{p}_{\text{p}}}\!\) |
|
|
\((((((~))(~))))\) |
| \(9\!\) |
\(\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\) |
\(\text{p}_\text{p}^\text{p}\!\) |
|
|
\((((~))(((~))))\) |
| \(16\!\) |
\(\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[6pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}\) |
\(\text{p}^{\text{p}^{\text{p}}}\!\) |
|
|
\(((((((~))))))\) |
Nested Wiki Table
Large Scale
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Small Scale
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Old ASCII Version
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
TeX Array
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\(\begin{array}{l|l|r} k & P_k = \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \} = \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \} & |P_k| \\[10pt] 0 & \{ 1 \} & 1 \\ 1 & \{ 2 \} & 1 \\ 2 & \{ 3, 4 \} & 2 \\ 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 \\ 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 \end{array}\) |
JPEG
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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
Wiki + TeX + JPEG
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\(1\!\) \(a(1) ~=~ 0\) |
\(\text{p}\!\) \(a(2) ~=~ 1\) |
\(\text{p}_\text{p}\!\) \(a(3) ~=~ 2\) |
\(\text{p}^\text{p}\!\) \(a(4) ~=~ 2\) |
\(\text{p}_{\text{p}_{\text{p}}}\!\) \(a(5) ~=~ 3\) |
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\(\text{p} \text{p}_{\text{p}}\!\) \(a(6) ~=~ 3\) |
\(\text{p}_{\text{p}^{\text{p}}}\!\) \(a(7) ~=~ 3\) |
\(\text{p}^{\text{p}_{\text{p}}}\!\) \(a(8) ~=~ 3\) |
\(\text{p}_\text{p}^\text{p}\!\) \(a(9) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(10) ~=~ 4\) |
|
\(\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!\) \(a(11) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(a(12) ~=~ 4\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}}}\!\) \(a(13) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p}^{\text{p}}}\!\) \(a(14) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(15) ~=~ 5\) |
|
\(\text{p}^{\text{p}^{\text{p}}}\!\) \(a(16) ~=~ 3\) |
\(\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!\) \(a(17) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(18) ~=~ 4\) |
\(\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!\) \(a(19) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(20) ~=~ 5\) |
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\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(21) ~=~ 5\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(22) ~=~ 5\) |
\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(23) ~=~ 4\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(a(24) ~=~ 5\) |
\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(25) ~=~ 4\) |
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\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(26) ~=~ 5\) |
\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(27) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(28) ~=~ 5\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(29) ~=~ 5\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(30) ~=~ 6\) |
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\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(a(31) ~=~ 5\) |
\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(a(32) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(33) ~=~ 6\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(34) ~=~ 5\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(35) ~=~ 6\) |
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\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(36) ~=~ 5\) |
\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(a(37) ~=~ 5\) |
\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(38) ~=~ 5\) |
\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(39) ~=~ 6\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(a(40) ~=~ 6\) |
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\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(a(41) ~=~ 5\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(42) ~=~ 6\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(a(43) ~=~ 5\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(44) ~=~ 6\) |
\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(45) ~=~ 6\) |
|
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(46) ~=~ 5\) |
\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(47) ~=~ 6\) |
\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(a(48) ~=~ 5\) |
\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(a(49) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(50) ~=~ 5\) |
|
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(51) ~=~ 6\) |
\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(52) ~=~ 6\) |
\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(a(53) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(54) ~=~ 5\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(55) ~=~ 7\) |
|
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(56) ~=~ 6\) |
\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(57) ~=~ 6\) |
\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(58) ~=~ 6\) |
\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(a(59) ~=~ 5\) |
\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(60) ~=~ 7\) |
A062860
Wiki + TeX + JPEG
|
\(1\!\) \(a(0) ~=~ 1\) |
\(\text{p}\!\) \(a(1) ~=~ 2\) |
\(\text{p}_\text{p}\!\) \(a(2) ~=~ 3\) |
\(\text{p}_{\text{p}_{\text{p}}}\!\) \(a(3) ~=~ 5\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(4) ~=~ 10\) |
|
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!\) \(a(5) ~=~ 15\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(6) ~=~ 30\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(7) ~=~ 55\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(8) ~=~ 105\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(9) ~=~ 165\) |
A106177
Primal Codes of Finite Partial Functions on Positive Integers
|
\(\begin{array}{rcl} 1 & = & \varnothing \\ 2 & = & 1\!:\!1 \\ 3 & = & 2\!:\!1 \\ 4 & = & 1\!:\!2 \\ 5 & = & 3\!:\!1 \\ 6 & = & 1\!:\!1 ~~ 2\!:\!1 \\ 7 & = & 4\!:\!1 \\ 8 & = & 1\!:\!3 \\ 9 & = & 2\!:\!2 \\ 10 & = & 1\!:\!1 ~~ 3\!:\!1 \\ 11 & = & 5\!:\!1 \\ 12 & = & 1\!:\!2 ~~ 2\!:\!1 \\ 13 & = & 6\!:\!1 \\ 14 & = & 1\!:\!1 ~~ 4\!:\!1 \\ 15 & = & 2\!:\!1 ~~ 3\!:\!1 \\ 16 & = & 1\!:\!4 \\ 17 & = & 7\!:\!1 \\ 18 & = & 1\!:\!1 ~~ 2\!:\!2 \\ 19 & = & 8\!:\!1 \\ 20 & = & 1\!:\!2 ~~ 3\!:\!1 \end{array}\) |
Wiki Table
| 1 | 1 | |||||||||||||||||||
| 2 | 1 | 2 | ||||||||||||||||||
| 3 | 1 | 1 | 3 | |||||||||||||||||
| 4 | 1 | 2 | 1 | 4 | ||||||||||||||||
| 5 | 1 | 3 | 1 | 1 | 5 | |||||||||||||||
| 6 | 1 | 1 | 1 | 4 | 1 | 6 | ||||||||||||||
| 7 | 1 | 5 | 2 | 9 | 1 | 1 | 7 | |||||||||||||
| 8 | 1 | 6 | 1 | 1 | 1 | 2 | 1 | 8 | ||||||||||||
| 9 | 1 | 7 | 1 | 25 | 1 | 3 | 1 | 1 | 9 | |||||||||||
| 10 | 1 | 1 | 1 | 36 | 1 | 2 | 1 | 8 | 1 | 10 |
Wiki + TeX
Smallmatrix
|
\(\begin{smallmatrix} & & & & & & & & & {\color{red}1} & & {\color{red}1} \\ & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} \\ & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} \\ & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} \\ & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} \\ & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} \\ & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} \\ & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} \\ & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} \\ {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} \end{smallmatrix}\) |
Array
|
\(\begin{array}{*{21}{c}} & & & & & & & & & {\color{red}1} & & {\color{red}1} \\ & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} \\ & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} \\ & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} \\ & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} \\ & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} \\ & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} \\ & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} \\ & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} \\ {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} \end{array}\) |
Matrix
|
\(\begin{matrix} n \circ m \\ 1 ~/~\backslash~ 1 \\ 2 ~/~ 1 ~\backslash~ 2 \\ 3 ~/~ 1 \cdot 1 ~\backslash~ 3 \\ 4 ~/~ 1 \cdot 2 \cdot 1 ~\backslash~ 4 \\ 5 ~/~ 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 5 \\ 6 ~/~ 1 \cdot 1 \cdot 1 \cdot 4 \cdot 1 ~\backslash~ 6 \\ 7 ~/~ 1 \cdot 5 \cdot 2 \cdot 9 \cdot 1 \cdot 1 ~\backslash~ 7 \\ 8 ~/~ 1 \cdot 6 \cdot 1 \cdot 1 \cdot 1 \cdot 2 \cdot 1 ~\backslash~ 8 \\ 9 ~/~ 1 \cdot 7 \cdot 1 \cdot 25\cdot 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 9 \\ 10 ~/~ 1 \cdot 1 \cdot 1 \cdot 36\cdot 1 \cdot 2 \cdot 1 \cdot 8 \cdot 1 ~\backslash~ 10 \end{matrix}\) |
ASCII
Example
* n o m
* \ /
* 1 . 1
* \ / \ /
* 2 . 1 . 2
* \ / \ / \ /
* 3 . 1 . 1 . 3
* \ / \ / \ / \ /
* 4 . 1 . 2 . 1 . 4
* \ / \ / \ / \ / \ /
* 5 . 1 . 3 . 1 . 1 . 5
* \ / \ / \ / \ / \ / \ /
* 6 . 1 . 1 . 1 . 4 . 1 . 6
* \ / \ / \ / \ / \ / \ / \ /
* 7 . 1 . 5 . 2 . 9 . 1 . 1 . 7
* \ / \ / \ / \ / \ / \ / \ / \ /
* 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8
* \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10
*
* Primal codes of finite partial functions on positive integers:
* 1 = { }
* 2 = 1:1
* 3 = 2:1
* 4 = 1:2
* 5 = 3:1
* 6 = 1:1 2:1
* 7 = 4:1
* 8 = 1:3
* 9 = 2:2
* 10 = 1:1 3:1
* 11 = 5:1
* 12 = 1:2 2:1
* 13 = 6:1
* 14 = 1:1 4:1
* 15 = 2:1 3:1
* 16 = 1:4
* 17 = 7:1
* 18 = 1:1 2:2
* 19 = 8:1
* 20 = 1:2 3:1
A106178
Wiki Table
| 1 | 1 | |||||||||||||||||||||||||||||||
| 2 | · | 2 | ||||||||||||||||||||||||||||||
| 3 | · | · | 3 | |||||||||||||||||||||||||||||
| 4 | · | 2 | · | 4 | ||||||||||||||||||||||||||||
| 5 | · | 3 | 1 | · | 5 | |||||||||||||||||||||||||||
| 6 | · | 1 | 1 | 4 | · | 6 | ||||||||||||||||||||||||||
| 7 | · | 5 | 2 | 9 | 1 | · | 7 | |||||||||||||||||||||||||
| 8 | · | 6 | 1 | 1 | 1 | 2 | · | 8 | ||||||||||||||||||||||||
| 9 | · | 7 | 1 | 25 | 1 | 3 | 1 | · | 9 | |||||||||||||||||||||||
| 10 | · | 1 | 1 | 36 | 1 | 2 | 1 | 8 | · | 10 | ||||||||||||||||||||||
| 11 | · | 1 | 1 | 49 | 1 | 5 | 1 | 27 | 1 | · | 11 | |||||||||||||||||||||
| 12 | · | 10 | 3 | 1 | 1 | 6 | 1 | 1 | 1 | 2 | · | 12 | ||||||||||||||||||||
| 13 | · | 11 | 1 | 1 | 2 | 7 | 1 | 125 | 4 | 3 | 1 | · | 13 | |||||||||||||||||||
| 14 | · | 3 | 1 | 100 | 1 | 1 | 1 | 216 | 1 | 1 | 1 | 4 | · | 14 | ||||||||||||||||||
| 15 | · | 13 | 2 | 121 | 1 | 3 | 1 | 343 | 1 | 5 | 1 | 9 | 1 | · | 15 | |||||||||||||||||
| 16 | · | 14 | 1 | 9 | 1 | 10 | 1 | 1 | 1 | 6 | 1 | 2 | 1 | 2 | · | 16 |
TeX Smallmatrix
|
\(\begin{smallmatrix} &&&&&&&&&&&&&&& {\color{red}1} && {\color{red}1} \\ &&&&&&&&&&&&&& {\color{red}2} && \cdot & & {\color{red}2} \\ &&&&&&&&&&&&& {\color{red}3} && \cdot && \cdot && {\color{red}3} \\ &&&&&&&&&&&& {\color{red}4} && \cdot && 2 && \cdot && {\color{red}4} \\ &&&&&&&&&&& {\color{red}5} && \cdot && 3 && 1 && \cdot && {\color{red}5} \\ &&&&&&&&&& {\color{red}6} && \cdot && 1 && 1 && 4 && \cdot && {\color{red}6} \\ &&&&&&&&& {\color{red}7} && \cdot && 5 && 2 && 9 && 1 && \cdot && {\color{red}7} \\ &&&&&&&& {\color{red}8} && \cdot && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}8} \\ &&&&&&& {\color{red}9} && \cdot && 7 && 1 && 25 && 1 && 3 && 1 && \cdot && {\color{red}9} \\ &&&&&& {\color{red}10} && \cdot && 1 && 1 && 36 && 1 && 2 && 1 && 8 && \cdot && {\color{red}10} \\ &&&&& {\color{red}11} && \cdot && 1 && 1 && 49 && 1 && 5 && 1 && 27 && 1 && \cdot && {\color{red}11} \\ &&&& {\color{red}12} && \cdot && 10 && 3 && 1 && 1 && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}12} \\ &&& {\color{red}13} && \cdot && 11 && 1 && 1 && 2 && 7 && 1 && 125 && 4 && 3 && 1 && \cdot && {\color{red}13} \\ && {\color{red}14} && \cdot && 3 && 1 && 100 && 1 && 1 && 1 && 216 && 1 && 1 && 1 && 4 && \cdot && {\color{red}14} \\ & {\color{red}15} && \cdot && 13 && 2 && 121 && 1 && 3 && 1 && 343 && 1 && 5 && 1 && 9 && 1 && \cdot && {\color{red}15} \\ {\color{red}16} && \cdot && 14 && 1 && 9 && 1 && 10 && 1 && 1 && 1 && 6 && 1 && 2 && 1 && 2 && \cdot && {\color{red}16} \end{smallmatrix}\) |
ASCII
Example
* n o m
* \ /
* 1 . 1
* \ / \ /
* 2 . . 2
* \ / \ / \ /
* 3 . . . 3
* \ / \ / \ / \ /
* 4 . . 2 . . 4
* \ / \ / \ / \ / \ /
* 5 . . 3 . 1 . . 5
* \ / \ / \ / \ / \ / \ /
* 6 . . 1 . 1 . 4 . . 6
* \ / \ / \ / \ / \ / \ / \ /
* 7 . . 5 . 2 . 9 . 1 . . 7
* \ / \ / \ / \ / \ / \ / \ / \ /
* 8 . . 6 . 1 . 1 . 1 . 2 . . 8
* \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 9 . . 7 . 1 . 25. 1 . 3 . 1 . . 9
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 10 . . 1 . 1 . 36. 1 . 2 . 1 . 8 . . 10
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 11 . . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 . . 11
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 12 . . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . . 12
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 13 . . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 . . 13
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 14 . . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 . . 14
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 15 . . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 . . 15
* \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
* 16 . . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 . . 16
A108352
Links
- Jon Awbrey, Primal Code Characteristic, n = 1 to 1000
- Jon Awbrey, Primal Code Characteristic, n = 1001 to 2000
- Jon Awbrey, Primal Code Characteristic, n = 2001 to 3000
TeX Array
|
\(\begin{array}{*{10}{l}} a(1) & = & 1 & \text{because} & (\circ~ 1)^1 & = & (\circ~ \varnothing)^1 & = & 1. \\ a(2) & = & 0 & \text{because} & (\circ~ 2)^k & = & (\circ~ 1\!:\!1)^k & = & 2, & \text{for all}~ k > 0. \\ a(3) & = & 2 & \text{because} & (\circ~ 3)^2 & = & (\circ~ 2\!:\!1)^2 & = & 1. \\ a(4) & = & 2 & \text{because} & (\circ~ 4 )^2 & = & (\circ~ 1\!:\!2)^2 & = &1. \\ a(5) & = & 2 & \text{because} & (\circ~ 5)^2 & = & (\circ~ 3\!:\!1)^2 & = & 1. \\ a(6) & = & 0 & \text{because} & (\circ~ 6)^k & = & (\circ~ 1\!:\!1 ~~ 2\!:\!1)^k & = & 6, & \text{for all}~ k > 0. \\ a(7) & = & 2 & \text{because} & (\circ~ 7)^2 & = & (\circ~ 4\!:\!1)^1 & = & 1. \\ a(8) & = & 2 & \text{because} & (\circ~ 8)^2 & = & (\circ~ 1\!:\!3)^1 & = & 1. \\ a(9) & = & 0 & \text{because} & (\circ~ 9)^k & = & (\circ~ 2\!:\!2)^k & = & 9, & \text{for all}~ k > 0. \\ a(10) & = & 0 & \text{because} & (\circ~ 10)^k & = & (\circ~ 1\!:\!1 ~~ 3\!:\!1)^k & = & 10, & \text{for all}~ k > 0. \end{array}\) |
ASCII
Example
* a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1.
* a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k.
* a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1.
* a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1.
* a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1.
* a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
* a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1.
* a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1.
* a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k.
* a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.
* Detail of calculation for compositional powers of 12:
* (12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
* (12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12
* Detail of calculation for compositional powers of 20:
* (20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
* (20 o)^3 = (3:2) o (1:2 3:1) = 1
A108371
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
A109300
JPEG
|
\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\) |
\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\) |
ASCII
Example
* Table of Rotes and Primal Functions for Positive Integers of Rote Height 2
*
* o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | | | | | |
* o-o o-o o-o o-o o---o o-o o-o o-o o---o o-o o---o
* | | | | | | | | | | |
* O O O===O O O=====O O===O O=====O
*
* 2:1 1:2 1:1 2:1 2:2 1:2 2:1 1:1 2:2 1:2 2:2
*
* 3 4 6 9 12 18 36
*
A109301
Example
\(802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1\)
\(\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}\)
\(\begin{array}{lllll} 802701 & = & 9 \cdot 89189 & = & 2\!:\!2 ~~ 8638\!:\!1 \\ 8638 & = & 2 \cdot 7 \cdot 617 & = & 1\!:\!1 ~~ 4\!:\!1 ~~ 113\!:\!1 \\ 113 & & & = & 30\!:\!1 \\ 30 & = & 2 \cdot 3 \cdot 5 & = & 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 4 & & & = & 1\!:\!2 \\ 3 & & & = & 2\!:\!1 \\ 2 & & & = & 1\!:\!1 \end{array}\)
\(\text{So the rote of 802701 is the following graph:}\)
\(\text{By inspection, the rote height of 802701 is 6.}\)
JPEG
|
\(1\!\) \(a(1) ~=~ 0\) |
\(\text{p}\!\) \(a(2) ~=~ 1\) |
\(\text{p}_\text{p}\!\) \(a(3) ~=~ 2\) |
\(\text{p}^\text{p}\!\) \(a(4) ~=~ 2\) |
\(\text{p}_{\text{p}_\text{p}}\!\) \(a(5) ~=~ 3\) |
|
\(\text{p} \text{p}_\text{p}\!\) \(a(6) ~=~ 2\) |
\(\text{p}_{\text{p}^\text{p}}\!\) \(a(7) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}}\!\) \(a(8) ~=~ 3\) |
\(\text{p}_\text{p}^\text{p}\!\) \(a(9) ~=~ 2\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(10) ~=~ 3\) |
|
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(11) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(a(12) ~=~ 2\) |
\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(13) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(14) ~=~ 3\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(15) ~=~ 3\) |
|
\(\text{p}^{\text{p}^\text{p}}\!\) \(a(16) ~=~ 3\) |
\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(17) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(18) ~=~ 2\) |
\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(19) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(20) ~=~ 3\) |
|
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(21) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(22) ~=~ 4\) |
\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(23) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(a(24) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(25) ~=~ 3\) |
|
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(26) ~=~ 3\) |
\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(27) ~=~ 3\) |
\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(28) ~=~ 3\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(29) ~=~ 4\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(30) ~=~ 3\) |
|
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(a(31) ~=~ 5\) |
\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(a(32) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(33) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(34) ~=~ 4\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(35) ~=~ 3\) |
|
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(36) ~=~ 2\) |
\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(a(37) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(38) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(39) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(a(40) ~=~ 3\) |
|
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(a(41) ~=~ 4\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(42) ~=~ 3\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(a(43) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(44) ~=~ 4\) |
\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(45) ~=~ 3\) |
|
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(46) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(47) ~=~ 4\) |
\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(a(48) ~=~ 3\) |
\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(a(49) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(50) ~=~ 3\) |
|
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(51) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(52) ~=~ 3\) |
\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(a(53) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(54) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(55) ~=~ 4\) |
|
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(56) ~=~ 3\) |
\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(57) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(58) ~=~ 4\) |
\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(a(59) ~=~ 5\) |
\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(60) ~=~ 3\) |
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.
