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Revision as of 18:48, 31 January 2010
, 18:48, 31 January 2010Undo revision 107256 by Jon Awbrey (Talk)
{| 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>
{| 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>
{| 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"
|
|
{| 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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<p> </p><br>
<p> </p><br>
<p> </p><br>
<p><math>1\!</math></p><br>
<p><math>a(1) ~=~ 0</math></p>
<p><math>a(1) ~=~ 0</math></p>
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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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| valign="bottom" |
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
<p> </p><br>
<p><math>1\!</math></p><br>
<p><math>a(1) ~=~ 0</math></p>
<p><math>a(1) ~=~ 0</math></p>
| valign="bottom" |
| valign="bottom" |
* 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>
