Changes

MyWikiBiz, Author Your Legacy — Friday November 01, 2024
Jump to navigationJump to search

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

Revision as of 22:00, 30 January 2016

2,641 bytes added ,  22:00, 30 January 2016
update
Line 1: Line 1:  
{{DISPLAYTITLE:Riffs and Rotes}}
 
{{DISPLAYTITLE:Riffs and Rotes}}
−
__TOC__
+
<div class="nonumtoc">__TOC__</div>
    
==Idea==
 
==Idea==
   −
Let <math>\text{p}_i</math> be the <math>i^\text{th}</math> prime, where the positive integer <math>i</math> is called the ''index'' of the prime  <math>\text{p}_i</math> and the indices are taken in such a way that <math>\text{p}_1 = 2.</math>  Thus the sequence of primes begins as follows:
+
Let <math>\text{p}_i\!</math> be the <math>i^\text{th}\!</math> prime, where the positive integer <math>i\!</math> is called the ''index'' of the prime  <math>\text{p}_i\!</math> and the indices are taken in such a way that <math>\text{p}_1 = 2.\!</math>  Thus the sequence of primes begins as follows:
    
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
Line 21: Line 21:  
|}
 
|}
   −
The prime factorization of a positive integer <math>n</math> can be written in the following form:
+
The prime factorization of a positive integer <math>n\!</math> can be written in the following form:
    
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
−
| <math>n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},</math>
+
| <math>n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},\!</math>
 
|}
 
|}
   −
where <math>\text{p}_{i(k)}^{j(k)}</math> is the <math>k^\text{th}</math> prime power in the factorization and <math>\ell</math> is the number of distinct prime factors dividing <math>n.</math>  The factorization of <math>1</math> is defined as <math>1</math> in accord with the convention that an empty product is equal to <math>1.</math>
+
where <math>\text{p}_{i(k)}^{j(k)}\!</math> is the <math>k^\text{th}\!</math> prime power in the factorization and <math>\ell\!</math> is the number of distinct prime factors dividing <math>n.\!</math>  The factorization of <math>1\!</math> is defined as <math>1\!</math> in accord with the convention that an empty product is equal to <math>1.\!</math>
   −
Let <math>I(n)</math> be the set of indices of primes that divide  <math>n</math> and let <math>j(i, n)</math> be the number of times that <math>\text{p}_i</math> divides <math>n.</math>  Then the prime factorization of <math>n</math> can be written in the following alternative form:
+
Let <math>I(n)\!</math> be the set of indices of primes that divide  <math>n\!</math> and let <math>j(i, n)\!</math> be the number of times that <math>\text{p}_i\!</math> divides <math>n.\!</math>  Then the prime factorization of <math>n\!</math> can be written in the following alternative form:
    
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
−
| <math>n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.</math>
+
| <math>n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.\!</math>
 
|}
 
|}
   Line 40: Line 40:  
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
9876543210
+
123456789
−
& = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721
+
& = & 3^2 \cdot 3607 \cdot 3803
−
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1.
+
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1.
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|}
 
|}
   −
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>32277</math> has the factorization <math>3 \cdot 7 \cdot 29 \cdot 53 = \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1.</math>  Taking this information together with previously known factorizations allows the following replacements to be made:
+
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%"
Line 55: Line 55:  
2 & \mapsto & \text{p}_1^1
 
2 & \mapsto & \text{p}_1^1
 
\\[6pt]
 
\\[6pt]
−
3 & \mapsto & \text{p}_2^1
+
504 & \mapsto & \text{p}_1^3 \text{p}_2^2 \text{p}_4^1
 
\\[6pt]
 
\\[6pt]
−
7 & \mapsto & \text{p}_4^1
+
529 & \mapsto & \text{p}_9^2
−
\\[6pt]
  −
32277 & \mapsto & \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1
   
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
Line 68: Line 66:  
|
 
|
 
<math>\begin{array}{lll}
 
<math>\begin{array}{lll}
−
9876543210
+
123456789
−
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1
+
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1
 
\\[12pt]
 
\\[12pt]
−
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1}^1 \text{p}_{\text{p}_4^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1}^1
+
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
   −
Continuing to replace every index and exponent with its factorization until no indices or exponents greater than <math>1</math> are left produces the following developemnt:
+
Continuing to replace every index and exponent with its factorization produces the following development:
    
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
 
|
 
|
 
<math>\begin{array}{lll}
 
<math>\begin{array}{lll}
−
9876543210
+
123456789
−
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1
+
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1
 
\\[18pt]
 
\\[18pt]
−
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1}^1 \text{p}_{\text{p}_4^1}^{\text{p}_1^1} \text{p}_{\text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1}^1
+
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1
 
\\[18pt]
 
\\[18pt]
−
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^2}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^2}^1 \text{p}_{\text{p}_1^1 \text{p}_3^1}^1 \text{p}_{\text{p}_1^4}^1}^1
+
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_2^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^2}^1}^1 \text{p}_{\text{p}_{\text{p}_2^2}^{\text{p}_1^1}}^1
 
\\[18pt]
 
\\[18pt]
−
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_2^1}^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^2}}^1}^1
+
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^{\text{p}_1^1}}^1
−
\\[18pt]
+
\end{array}</math>
−
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^{\text{p}_1^1} \text{p}_{\text{p}_{\text{p}_1^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1}^1
+
|}
 +
 
 +
The <math>1\!</math>'s that appear as indices and exponents are formally redundant, conveying no information apart from the places they occupy in the resulting syntactic structure.  Leaving them tacit produces the following expression:
 +
 
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{lll}
 +
123456789
 +
& = & \text{p}_{\text{p}}^{\text{p}} \text{p}_{\text{p}^{\text{p}_{\text{p}}} \text{p}_{\text{p}}^{\text{p}} \text{p}_{\text{p}^{\text{p}}}} \text{p}_{\text{p}_{\text{p}_{\text{p}}^{\text{p}}}^{\text{p}}}
 
\end{array}</math>
 
\end{array}</math>
 +
|}
 +
 +
The pattern of indices and exponents illustrated here is called a ''doubly recursive factorization'', or ''DRF''.  Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the DRF of <math>n.\!</math> &nbsp; If <math>\mathbb{M}</math> is the set of positive integers, <math>\mathcal{L}</math> is the set of DRF expressions, and the mapping defined by the factorization process is denoted <math>\operatorname{drf} : \mathbb{M} \to \mathcal{L},</math> then the doubly recursive factorization of <math>n\!</math> is denoted <math>\operatorname{drf}(n).\!</math>
 +
 +
The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called ''riffs'' and ''rotes'', respectively.
 +
 +
{| align=center cellpadding="6" width="90%"
 +
|-
 +
| <math>\operatorname{riff}(123456789)</math> is the following digraph:
 +
|-
 +
| align=center | [[Image:Riff 123456789 Big.jpg|220px]]
 +
|-
 +
| <math>\operatorname{rote}(123456789)</math> is the following graph:
 +
|-
 +
| align=center | [[Image:Rote 123456789 Big.jpg|345px]]
 
|}
 
|}
    
==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" |
Line 353: Line 374:     
{| 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>
Line 604: Line 626:  
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}</math></p>
 
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}</math></p>
 +
|}
 +
 +
==Prime Animations==
 +
 +
===Riffs 1 to 60===
 +
 +
{| align="center"
 +
| [[Image:Animation Riff 60 x 0.16.gif]]
 +
|}
 +
 +
===Rotes 1 to 60===
 +
 +
{| align="center"
 +
| [[Image:Animation Rote 60 x 0.16.gif]]
 
|}
 
|}
   Line 614: Line 650:  
* '''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%"
Line 754: Line 790:  
* '''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"
Line 1,159: Line 1,195:  
* '''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"
Line 1,420: Line 1,456:  
* '''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"
Line 1,471: Line 1,507:  
* '''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
Line 1,772: Line 1,808:  
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>a(60) ~=~ 3</math></p>
 
<p><math>a(60) ~=~ 3</math></p>
 +
|}
 +
 +
==Miscellaneous Examples==
 +
 +
{| align="center" border="1" width="96%"
 +
|+ style="height:24px" | <math>\text{Integers, Riffs, Rotes}\!</math>
 +
|- style="height:50px; background:#f0f0ff"
 +
|
 +
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
 +
| width="10%" | <math>\text{Integer}\!</math>
 +
| width="45%" | <math>\text{Riff}\!</math>
 +
| width="45%" | <math>\text{Rote}\!</math>
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>1\!</math>
 +
| width="45%" | &nbsp;
 +
| width="45%" | [[Image:Rote 1 Big.jpg|15px]]
 +
|-
 +
| <math>2\!</math>
 +
| [[Image:Riff 2 Big.jpg|15px]]
 +
| [[Image:Rote 2 Big.jpg|30px]]
 +
|-
 +
| <math>3\!</math>
 +
| [[Image:Riff 3 Big.jpg|30px]]
 +
| [[Image:Rote 3 Big.jpg|30px]]
 +
|-
 +
| <math>4\!</math>
 +
| [[Image:Riff 4 Big.jpg|30px]]
 +
| [[Image:Rote 4 Big.jpg|48px]]
 +
|-
 +
| <math>360\!</math>
 +
| [[Image:Riff 360 Big.jpg|120px]]
 +
| [[Image:Rote 360 Big.jpg|135px]]
 +
|-
 +
| <math>2010\!</math>
 +
| [[Image:Riff 2010 Big.jpg|138px]]
 +
| [[Image:Rote 2010 Big.jpg|144px]]
 +
|-
 +
| <math>2011\!</math>
 +
| [[Image:Riff 2011 Big.jpg|84px]]
 +
| [[Image:Rote 2011 Big.jpg|120px]]
 +
|-
 +
| <math>2012\!</math>
 +
| [[Image:Riff 2012 Big.jpg|100px]]
 +
| [[Image:Rote 2012 Big.jpg|125px]]
 +
|-
 +
| <math>2500\!</math>
 +
| [[Image:Riff 2500 Big.jpg|66px]]
 +
| [[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

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/107841...467174"

Navigation menu