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Directory:Jon Awbrey/Papers/Riffs and Rotes (view source)
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{{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%"
|}
|}
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>
|}
|}
|
|
<math>\begin{matrix}
<math>\begin{matrix}
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%"
2 & \mapsto & \text{p}_1^1
2 & \mapsto & \text{p}_1^1
\\[6pt]
\\[6pt]
504 & \mapsto & \text{p}_1^3 \text{p}_2^2 \text{p}_4^1
\\[6pt]
\\[6pt]
529 & \mapsto & \text{p}_9^2
\end{array}</math>
\end{array}</math>
|}
|}
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
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}
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
\end{array}</math>
|}
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> 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" |
{| 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>
<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]]
|}
|}
* '''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%"
* '''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"
* '''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"
* '''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"
* '''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
<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%" |
| 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]]
|}
|}
|}