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Undo revision 107256 by Jon Awbrey (Talk)
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{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
|+ style="height:24px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:48px; background:#f0f0ff"
 
|- style="height:48px; background:#f0f0ff"
 
| <math>\text{Integer}\!</math>
 
| <math>\text{Integer}\!</math>
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{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
|+ style="height:24px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:48px; background:#f0f0ff"
 
|- style="height:48px; background:#f0f0ff"
 
| <math>\text{Integer}\!</math>
 
| <math>\text{Integer}\!</math>
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{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:50px; background:#f0f0ff"
 
|- style="height:50px; background:#f0f0ff"
 
|
 
|
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{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
+
|+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math>
 
|- style="height:50px; background:#f0f0ff"
 
|- style="height:50px; background:#f0f0ff"
 
|
 
|
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* [http://oeis.org/wiki/A109301 A109301]
 
* [http://oeis.org/wiki/A109301 A109301]
 +
 +
===Example===
 +
 +
: <math>802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1</math>
 +
 +
: <math>\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}</math>
 +
 +
: <math>\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}</math>
 +
 +
: <math>\text{So the rote of 802701 is the following graph:}\!</math>
 +
 +
:{| border="1" cellpadding="20"
 +
| [[Image:Rote 802701 Big.jpg|330px]]
 +
|}
 +
 +
: <math>\text{By inspection, the rote height of 802701 is 6.}\!</math>
    
===JPEG===
 
===JPEG===
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     * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
 
     * 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
 
     * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
     * 7.840858554516122655953405327738 x 10^371.  
+
     * 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.
 
</pre>
 
</pre>
  
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