Difference between revisions of "Directory:Jon Awbrey/Papers/Riffs and Rotes"

MyWikiBiz, Author Your Legacy — Tuesday May 20, 2025
Jump to navigationJump to search
Line 60: Line 60:
 
\\[6pt]
 
\\[6pt]
 
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

Rote 1 Big.jpg


1\!


a(1) ~=~ 0

Rote 2 Big.jpg


\text{p}\!


a(2) ~=~ 1

Rote 3 Big.jpg


\text{p}_\text{p}\!


a(3) ~=~ 2

Rote 4 Big.jpg


\text{p}^\text{p}\!


a(4) ~=~ 2

Rote 5 Big.jpg


\text{p}_{\text{p}_\text{p}}\!


a(5) ~=~ 3

Rote 6 Big.jpg


\text{p} \text{p}_\text{p}\!


a(6) ~=~ 2

Rote 7 Big.jpg


\text{p}_{\text{p}^\text{p}}\!


a(7) ~=~ 3

Rote 8 Big.jpg


\text{p}^{\text{p}_\text{p}}\!


a(8) ~=~ 3

Rote 9 Big.jpg


\text{p}_\text{p}^\text{p}\!


a(9) ~=~ 2

Rote 10 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}\!


a(10) ~=~ 3

Rote 11 Big.jpg


\text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(11) ~=~ 4

Rote 12 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}\!


a(12) ~=~ 2

Rote 13 Big.jpg


\text{p}_{\text{p} \text{p}_\text{p}}\!


a(13) ~=~ 3

Rote 14 Big.jpg


\text{p} \text{p}_{\text{p}^\text{p}}\!


a(14) ~=~ 3

Rote 15 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(15) ~=~ 3

Rote 16 Big.jpg


\text{p}^{\text{p}^\text{p}}\!


a(16) ~=~ 3

Rote 17 Big.jpg


\text{p}_{\text{p}_{\text{p}^\text{p}}}\!


a(17) ~=~ 4

Rote 18 Big.jpg


\text{p} \text{p}_\text{p}^\text{p}\!


a(18) ~=~ 2

Rote 19 Big.jpg


\text{p}_{\text{p}^{\text{p}_\text{p}}}\!


a(19) ~=~ 4

Rote 20 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


a(20) ~=~ 3

Rote 21 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!


a(21) ~=~ 3

Rote 22 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(22) ~=~ 4

Rote 23 Big.jpg


\text{p}_{\text{p}_\text{p}^\text{p}}\!


a(23) ~=~ 3

Rote 24 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!


a(24) ~=~ 3

Rote 25 Big.jpg


\text{p}_{\text{p}_\text{p}}^\text{p}\!


a(25) ~=~ 3

Rote 26 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(26) ~=~ 3

Rote 27 Big.jpg


\text{p}_\text{p}^{\text{p}_\text{p}}\!


a(27) ~=~ 3

Rote 28 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!


a(28) ~=~ 3

Rote 29 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(29) ~=~ 4

Rote 30 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(30) ~=~ 3

Rote 31 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!


a(31) ~=~ 5

Rote 32 Big.jpg


\text{p}^{\text{p}_{\text{p}_\text{p}}}\!


a(32) ~=~ 4

Rote 33 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(33) ~=~ 4

Rote 34 Big.jpg


\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


a(34) ~=~ 4

Rote 35 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


a(35) ~=~ 3

Rote 36 Big.jpg


\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!


a(36) ~=~ 2

Rote 37 Big.jpg


\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!


a(37) ~=~ 3

Rote 38 Big.jpg


\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


a(38) ~=~ 4

Rote 39 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(39) ~=~ 3

Rote 40 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!


a(40) ~=~ 3

Rote 41 Big.jpg


\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!


a(41) ~=~ 4

Rote 42 Big.jpg


\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!


a(42) ~=~ 3

Rote 43 Big.jpg


\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!


a(43) ~=~ 4

Rote 44 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(44) ~=~ 4

Rote 45 Big.jpg


\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!


a(45) ~=~ 3

Rote 46 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!


a(46) ~=~ 3

Rote 47 Big.jpg


\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(47) ~=~ 4

Rote 48 Big.jpg


\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!


a(48) ~=~ 3

Rote 49 Big.jpg


\text{p}_{\text{p}^\text{p}}^\text{p}\!


a(49) ~=~ 3

Rote 50 Big.jpg


\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!


a(50) ~=~ 3

Rote 51 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!


a(51) ~=~ 4

Rote 52 Big.jpg


\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!


a(52) ~=~ 3

Rote 53 Big.jpg


\text{p}_{\text{p}^{\text{p}^\text{p}}}\!


a(53) ~=~ 4

Rote 54 Big.jpg


\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!


a(54) ~=~ 3

Rote 55 Big.jpg


\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!


a(55) ~=~ 4

Rote 56 Big.jpg


\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!


a(56) ~=~ 3

Rote 57 Big.jpg


\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!


a(57) ~=~ 4

Rote 58 Big.jpg


\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!


a(58) ~=~ 4

Rote 59 Big.jpg


\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!


a(59) ~=~ 5

Rote 60 Big.jpg


\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!


a(60) ~=~ 3