Changes

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%"

| align=center | [[Image:Rote 123456789 Big.jpg|345px]]

| 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"

| valign="bottom" |

| valign="bottom" |

<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>

<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>

| [[Image:Animation Rote 60 x 0.16.gif]]

| [[Image:Animation Rote 60 x 0.16.gif]]

|}

|}

==Selected Sequences==

==Selected Sequences==

| [[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>2010\!</math>

| <math>2010\!</math>