Difference between revisions of "Directory:Jon Awbrey/Papers/Riffs and Rotes"
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32277 & \mapsto & \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1 | 32277 & \mapsto & \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1 | ||
+ | \end{array}</math> | ||
+ | |} | ||
+ | |||
+ | This leads to the following development: | ||
+ | |||
+ | {| align="center" cellpadding="6" width="90%" | ||
+ | | | ||
+ | <math>\begin{array}{lll} | ||
+ | 9876543210 | ||
+ | & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1 | ||
+ | \\[6pt] | ||
+ | & = & \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 | ||
\end{array}</math> | \end{array}</math> | ||
|} | |} |
Revision as of 20:20, 3 February 2010
Idea
Let be the i^\text{th} prime, where the positive integer i is called the index of the prime \text{p}_i and the indices are taken in such a way that \text{p}_1 = 2. Thus the sequence of primes begins as follows:
\begin{matrix} \text{p}_1 = 2, & \text{p}_2 = 3, & \text{p}_3 = 5, & \text{p}_4 = 7, & \text{p}_5 = 11, & \text{p}_6 = 13, & \text{p}_7 = 17, & \text{p}_8 = 19, & \ldots \end{matrix} |
The prime factorization of a positive integer n can be written in the following form:
n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)}, |
where \text{p}_{i(k)}^{j(k)} is the k^\text{th} prime power in the factorization and \ell is the number of distinct prime factors dividing n. The factorization of 1 is defined as 1 in accord with the convention that an empty product is equal to 1.
Let I(n) be the set of indices of primes that divide n and let j(i, n) be the number of times that \text{p}_i divides n. Then the prime factorization of n can be written in the following alternative form:
n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}. |
For example:
\begin{matrix} 9876543210 & = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721 & = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1. \end{matrix} |
Each index i and exponent j appearing in the prime factorization of a positive integer n is itself a positive integer, and thus has a prime factorization of its own.
Continuing with the same example, the index 32277 has the factorization 3 \cdot 7 \cdot 29 \cdot 53 = \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1. Taking this information together with previously known factorizations allows the following replacements to be made:
\begin{array}{rcl} 2 & \mapsto & \text{p}_1^1 \'"`UNIQ-MathJax1-QINU`"' '"`UNIQ-MathJax2-QINU`"' '"`UNIQ-MathJax3-QINU`"' '"`UNIQ-MathJax4-QINU`"' :{| border="1" cellpadding="20" | [[Image:Rote 802701 Big.jpg|330px]] |} '"`UNIQ-MathJax5-QINU`"' <br> {| align="center" border="1" cellpadding="6" |+ style="height:25px" | \(a(n) = \text{Rote Height of}~ n |
1\! a(1) ~=~ 0 |
\text{p}\! a(2) ~=~ 1 |
\text{p}_\text{p}\! a(3) ~=~ 2 |
\text{p}^\text{p}\! a(4) ~=~ 2 |
\text{p}_{\text{p}_\text{p}}\! a(5) ~=~ 3 |
\text{p} \text{p}_\text{p}\! a(6) ~=~ 2 |
\text{p}_{\text{p}^\text{p}}\! a(7) ~=~ 3 |
\text{p}^{\text{p}_\text{p}}\! a(8) ~=~ 3 |
\text{p}_\text{p}^\text{p}\! a(9) ~=~ 2 |
\text{p} \text{p}_{\text{p}_\text{p}}\! a(10) ~=~ 3 | |
\text{p}_{\text{p}_{\text{p}_\text{p}}}\! a(11) ~=~ 4 |
\text{p}^\text{p} \text{p}_\text{p}\! a(12) ~=~ 2 |
\text{p}_{\text{p} \text{p}_\text{p}}\! a(13) ~=~ 3 |
\text{p} \text{p}_{\text{p}^\text{p}}\! a(14) ~=~ 3 |
\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\! a(15) ~=~ 3 | |
\text{p}^{\text{p}^\text{p}}\! a(16) ~=~ 3 |
\text{p}_{\text{p}_{\text{p}^\text{p}}}\! a(17) ~=~ 4 |
\text{p} \text{p}_\text{p}^\text{p}\! a(18) ~=~ 2 |
\text{p}_{\text{p}^{\text{p}_\text{p}}}\! a(19) ~=~ 4 |
\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\! a(20) ~=~ 3 | |
\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\! a(21) ~=~ 3 |
\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\! a(22) ~=~ 4 |
\text{p}_{\text{p}_\text{p}^\text{p}}\! a(23) ~=~ 3 |
\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\! a(24) ~=~ 3 |
\text{p}_{\text{p}_\text{p}}^\text{p}\! a(25) ~=~ 3 | |
\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\! a(26) ~=~ 3 |
\text{p}_\text{p}^{\text{p}_\text{p}}\! a(27) ~=~ 3 |
\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\! a(28) ~=~ 3 |
\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\! a(29) ~=~ 4 |
\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\! a(30) ~=~ 3 | |
\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\! a(31) ~=~ 5 |
\text{p}^{\text{p}_{\text{p}_\text{p}}}\! a(32) ~=~ 4 |
\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\! a(33) ~=~ 4 |
\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\! a(34) ~=~ 4 |
\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\! a(35) ~=~ 3 | |
\text{p}^\text{p} \text{p}_\text{p}^\text{p}\! a(36) ~=~ 2 |
\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\! a(37) ~=~ 3 |
\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\! a(38) ~=~ 4 |
\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\! a(39) ~=~ 3 |
\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\! a(40) ~=~ 3 | |
\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\! a(41) ~=~ 4 |
\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\! a(42) ~=~ 3 |
\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\! a(43) ~=~ 4 |
\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\! a(44) ~=~ 4 |
\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\! a(45) ~=~ 3 | |
\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\! a(46) ~=~ 3 |
\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\! a(47) ~=~ 4 |
\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\! a(48) ~=~ 3 |
\text{p}_{\text{p}^\text{p}}^\text{p}\! a(49) ~=~ 3 |
\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\! a(50) ~=~ 3 | |
\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\! a(51) ~=~ 4 |
\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\! a(52) ~=~ 3 |
\text{p}_{\text{p}^{\text{p}^\text{p}}}\! a(53) ~=~ 4 |
\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\! a(54) ~=~ 3 |
\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\! a(55) ~=~ 4 | |
\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\! a(56) ~=~ 3 |
\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\! a(57) ~=~ 4 |
\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\! a(58) ~=~ 4 |
\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\! a(59) ~=~ 5 |
\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\! a(60) ~=~ 3 |