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Directory:Jon Awbrey/Papers/Riffs and Rotes (view source)

Revision as of 22:00, 30 January 2016

971 bytes removed ,  22:00, 30 January 2016
update
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Each index <math>i\!</math> and exponent <math>j\!</math> appearing in the prime factorization of a positive integer <math>n\!</math> is itself a positive integer, and thus has a prime factorization of its own.
 
Each index <math>i\!</math> and exponent <math>j\!</math> appearing in the prime factorization of a positive integer <math>n\!</math> is itself a positive integer, and thus has a prime factorization of its own.
   −
Continuing with the same example, the index <math>504\!</math> has the factorization <math>2^3 \cdot 3^2 \cdot 7 = \text{p}_1^3 \text{p}_2^2 \text{p}_4^1\!</math> and the index <math>529\!</math> has the factorization <math>{23}^2 = \text{p}_9^2.\!</math>  Taking this information together with previously known factorizations allows the following replacements to be made in the above expression:
+
Continuing with the same example, the index <math>504\!</math> has the factorization <math>2^3 \cdot 3^2 \cdot 7 = \text{p}_1^3 \text{p}_2^2 \text{p}_4^1\!</math> and the index <math>529\!</math> has the factorization <math>{23}^2 = \text{p}_9^2.\!</math>  Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:
    
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
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| align=center | [[Image:Rote 123456789 Big.jpg|345px]]
 
| align=center | [[Image:Rote 123456789 Big.jpg|345px]]
 
|}
 
|}
−
  −
==Definitions==
      
==Riffs in Numerical Order==
 
==Riffs in Numerical Order==
   −
{| align="center" border="1" cellpadding="10"
+
{| align="center" border="1" cellpadding="12"
 
|+ style="height:25px" | <math>\text{Riffs in Numerical Order}\!</math>
 
|+ style="height:25px" | <math>\text{Riffs in Numerical Order}\!</math>
 
| valign="bottom" |
 
| valign="bottom" |
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{| align="center" border="1" cellpadding="6"
 
{| align="center" border="1" cellpadding="6"
 +
|+ style="height:25px" | <math>\text{Rotes in Numerical Order}\!</math>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
 
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
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|}
 
|}
   −
==Prime Movies==
+
==Prime Animations==
   −
===Riffs===
+
===Riffs 1 to 60===
   −
<embed>
+
{| align="center"
−
<object width="200" height="200" ><param name="allowfullscreen" value="true" /><param name="allowscriptaccess" value="always" /><param name="movie" value="http://www.facebook.com/v/1641186383071" /><embed src="http://www.facebook.com/v/1641186383071" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="200" height="200">
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| [[Image:Animation Riff 60 x 0.16.gif]]
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</embed>
+
|}
   −
<embed>
+
===Rotes 1 to 60===
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<object width="200" height="200" ><param name="allowfullscreen" value="true" /><param name="allowscriptaccess" value="always" /><param name="movie" value="http://www.facebook.com/v/1641852119714" /><embed src="http://www.facebook.com/v/1641852119714" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="200" height="200">
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</embed>
     −
===Rotes===
+
{| align="center"
−
 
+
| [[Image:Animation Rote 60 x 0.16.gif]]
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<embed>
+
|}
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<object width="200" height="200" ><param name="allowfullscreen" value="true" /><param name="allowscriptaccess" value="always" /><param name="movie" value="http://www.facebook.com/v/1641177462848" /><embed src="http://www.facebook.com/v/1641177462848" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="200" height="200">
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</embed>
  −
 
  −
<embed>
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<object width="200" height="200" ><param name="allowfullscreen" value="true" /><param name="allowscriptaccess" value="always" /><param name="movie" value="http://www.facebook.com/v/1641847559600" /><embed src="http://www.facebook.com/v/1641847559600" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="200" height="200">
  −
</embed>
      
==Selected Sequences==
 
==Selected Sequences==
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* '''Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.'''
 
* '''Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.'''
   −
* [http://oeis.org/wiki/A061396 OEIS Wiki Entry for A061396].
+
* [http://oeis.org/A061396 OEIS Entry for A061396].
    
{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
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* '''Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.'''
 
* '''Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.'''
   −
* [http://oeis.org/wiki/A062504 OEIS Wiki Entry for A062504].
+
* [http://oeis.org/A062504 OEIS Entry for A062504].
    
{| align="center"
 
{| align="center"
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* '''Nodes in riff (rooted index-functional forest) for n.'''
 
* '''Nodes in riff (rooted index-functional forest) for n.'''
   −
* [http://oeis.org/wiki/A062537 OEIS Wiki Entry for A062537].
+
* [http://oeis.org/A062537 OEIS Entry for A062537].
    
{| align="center" border="1" cellpadding="10"
 
{| align="center" border="1" cellpadding="10"
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* '''Smallest j with n nodes in its riff (rooted index-functional forest).'''
 
* '''Smallest j with n nodes in its riff (rooted index-functional forest).'''
   −
* [http://oeis.org/wiki/A062860 OEIS Wiki Entry for A062860].
+
* [http://oeis.org/A062860 OEIS Entry for A062860].
    
{| align="center" border="1" cellpadding="10"
 
{| align="center" border="1" cellpadding="10"
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* '''a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.'''
 
* '''a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.'''
   −
* [http://oeis.org/wiki/A109301 OEIS Wiki Entry for A109301].
+
* [http://oeis.org/A109301 OEIS Entry for A109301].
    
; Example
 
; Example
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| [[Image:Riff 4 Big.jpg|30px]]
 
| [[Image:Riff 4 Big.jpg|30px]]
 
| [[Image:Rote 4 Big.jpg|48px]]
 
| [[Image:Rote 4 Big.jpg|48px]]
 +
|-
 +
| <math>360\!</math>
 +
| [[Image:Riff 360 Big.jpg|120px]]
 +
| [[Image:Rote 360 Big.jpg|135px]]
 
|-
 
|-
 
| <math>2010\!</math>
 
| <math>2010\!</math>
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| [[Image:Riff 2011 Big.jpg|84px]]
 
| [[Image:Riff 2011 Big.jpg|84px]]
 
| [[Image:Rote 2011 Big.jpg|120px]]
 
| [[Image:Rote 2011 Big.jpg|120px]]
 +
|-
 +
| <math>2012\!</math>
 +
| [[Image:Riff 2012 Big.jpg|100px]]
 +
| [[Image:Rote 2012 Big.jpg|125px]]
 
|-
 
|-
 
| <math>2500\!</math>
 
| <math>2500\!</math>
 
| [[Image:Riff 2500 Big.jpg|66px]]
 
| [[Image:Riff 2500 Big.jpg|66px]]
 
| [[Image:Rote 2500 Big.jpg|125px]]
 
| [[Image:Rote 2500 Big.jpg|125px]]
 +
|-
 +
| <math>802701\!</math>
 +
| [[Image:Riff 802701 Big.jpg|156px]]
 +
| [[Image:Rote 802701 Big.jpg|245px]]
 +
|-
 +
| <math>123456789\!</math>
 +
| [[Image:Riff 123456789 Big.jpg|162px]]
 +
| [[Image:Rote 123456789 Big.jpg|256px]]
 
|}
 
|}
 
|}
 
|}
Jon Awbrey
12,080

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