Directory:Jon Awbrey/Papers/Riffs and Rotes - Revision history

Jon Awbrey: update

2016-01-30T22:00:16Z

update

Jon Awbrey: add animations

2013-01-26T16:28:12Z

add animations

Jon Awbrey: /* Miscellaneous Examples */

2012-07-05T03:42:54Z

Miscellaneous Examples

Jon Awbrey: /* Miscellaneous Examples */

2012-07-05T03:40:17Z

Miscellaneous Examples

Jon Awbrey: /* Miscellaneous Examples */ + 123456789

2011-01-07T18:40:37Z

Miscellaneous Examples: + 123456789

Jon Awbrey: /* Miscellaneous Examples */ + 802701

2011-01-07T18:34:19Z

Miscellaneous Examples: + 802701

Jon Awbrey: /* A109301 */ update link

2011-01-05T20:06:13Z

A109301: update link

Jon Awbrey: /* A062860 */ update link

2011-01-05T20:04:11Z

A062860: update link

Jon Awbrey: /* A062537 */ update link

2011-01-05T20:02:19Z

A062537: update link

Jon Awbrey: /* A062504 */ update link

2011-01-05T20:00:13Z

A062504: update link

@@ Line 48: / Line 48: @@
 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%"
@@ Line 113: / Line 113: @@
 | align=center | [[Image:Rote 123456789 Big.jpg|345px]]
 |}
 ==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>
 | valign="bottom" |
@@ Line 376: / Line 374: @@
 {| align="center" border="1" cellpadding="6"
 | valign="bottom" |
 <p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
@@ Line 642: / Line 641: @@
 | [[Image:Animation Rote 60 x 0.16.gif]]
 |}
 ==Selected Sequences==
@@ Line 1,862: / Line 1,839: @@
 | [[Image:Riff 4 Big.jpg|30px]]
 | [[Image:Rote 4 Big.jpg|48px]]
 |-
 | <math>2010\!</math>

@@ Line 627: / Line 627: @@
 <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>
 |}

@@ Line 1,860: / Line 1,860: @@
 | [[Image:Riff 2012 Big.jpg|100px]]
 | [[Image:Rote 2012 Big.jpg|125px]]
 |-
 | <math>2500\!</math>

@@ Line 1,856: / Line 1,856: @@
 | [[Image:Riff 2011 Big.jpg|84px]]
 | [[Image:Rote 2011 Big.jpg|120px]]
 |-
 | <math>2500\!</math>

@@ Line 1,864: / Line 1,864: @@
 | [[Image:Riff 802701 Big.jpg|156px]]
 | [[Image:Rote 802701 Big.jpg|245px]]
 |}
 |}

@@ Line 1,860: / Line 1,860: @@
 | [[Image:Riff 2500 Big.jpg|66px]]
 | [[Image:Rote 2500 Big.jpg|125px]]
 |}
 |}

@@ Line 1,516: / Line 1,516: @@
 * '''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

@@ Line 1,465: / Line 1,465: @@
 * '''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"

@@ Line 1,204: / Line 1,204: @@
 * '''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"

@@ Line 799: / Line 799: @@
 * '''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"