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Riffs and Rotes

MyWikiBiz, Author Your Legacy — Tuesday March 26, 2019
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Idea

Let \(\text{p}_i\!\) 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:

\(\begin{matrix} 123456789 & = & 3^2 \cdot 3607 \cdot 3803 & = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1. \end{matrix}\)

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 \(504\!\) has the factorization \(2^3 \cdot 3^2 \cdot 7 = \text{p}_1^3 \text{p}_2^2 \text{p}_4^1\!\) and the index \(529\!\) has the factorization \({23}^2 = \text{p}_9^2.\!\) Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:

\(\begin{array}{rcl} 2 & \mapsto & \text{p}_1^1 \UNIQ03ebeeb9e3e11619-MathJax-1-QINU UNIQ03ebeeb9e3e11619-MathJax-2-QINU UNIQ03ebeeb9e3e11619-MathJax-3-QINU UNIQ03ebeeb9e3e11619-MathJax-4-QINU :{| border="1" cellpadding="20" | [[Image:Rote 802701 Big.jpg|330px]] |} UNIQ03ebeeb9e3e11619-MathJax-5-QINU
{| 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


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


\(a(60) ~=~ 3\)

Miscellaneous Examples

\(\text{Integers, Riffs, Rotes}\!\)
\(\text{Integer}\!\) \(\text{Riff}\!\) \(\text{Rote}\!\)
\(1\!\)   Rote 1 Big.jpg
\(2\!\) Riff 2 Big.jpg Rote 2 Big.jpg
\(3\!\) Riff 3 Big.jpg Rote 3 Big.jpg
\(4\!\) Riff 4 Big.jpg Rote 4 Big.jpg
\(360\!\) 120px 135px
\(2010\!\) Riff 2010 Big.jpg Rote 2010 Big.jpg
\(2011\!\) Riff 2011 Big.jpg Rote 2011 Big.jpg
\(2012\!\) Riff 2012 Big.jpg Rote 2012 Big.jpg
\(2500\!\) Riff 2500 Big.jpg Rote 2500 Big.jpg
\(802701\!\) Riff 802701 Big.jpg Rote 802701 Big.jpg
\(123456789\!\) Riff 123456789 Big.jpg Rote 123456789 Big.jpg