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Riffs and Rotes
From OeisWiki
Contents |
Idea
Let 
be the 
prime, where the positive integer 
is called the index of the prime and the indices are taken in such a way that 
Thus the sequence of primes begins as follows:
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The prime factorization of a positive integer 
can be written in the following form:
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where 
is the 
prime power in the factorization and 
is the number of distinct prime factors dividing 
The factorization of 
is defined as in accord with the convention that an empty product is equal to 
Let 
be the set of indices of primes that divide and let 
be the number of times that divides Then the prime factorization of can be written in the following alternative form:
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For example:
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Each index and exponent 
appearing in the prime factorization of a positive integer is itself a positive integer, and thus has a prime factorization of its own.
Continuing with the same example, the index 
has the factorization 
and the index 
has the factorization 
Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:
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![\begin{array}{rcl}
2 & \mapsto & \text{p}_1^1
\\[6pt]
504 & \mapsto & \text{p}_1^3 \text{p}_2^2 \text{p}_4^1
\\[6pt]
529 & \mapsto & \text{p}_9^2
\end{array}](/w/images/math/8/6/2/8620e159681b8b7c58d77f5892e5d5c3.png)
This leads to the following development:
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![\begin{array}{lll}
123456789
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1
\end{array}](/w/images/math/f/a/2/fa2e20e67e0e84e8ba3e386084c53fdf.png)
Continuing to replace every index and exponent with its factorization produces the following development:
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![\begin{array}{lll}
123456789
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1
\\[18pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1
\\[18pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_2^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^2}^1}^1 \text{p}_{\text{p}_{\text{p}_2^2}^{\text{p}_1^1}}^1
\\[18pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^{\text{p}_1^1}}^1
\end{array}](/w/images/math/e/6/5/e6527f3d8935ef5ea177d620e8f1ed9f.png)
The 's that appear as indices and exponents are formally redundant, conveying no information apart from the places they occupy in the resulting syntactic structure. Leaving them tacit produces the following expression:
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The pattern of indices and exponents illustrated here is called a doubly recursive factorization, or DRF. Applying the same procedure to any positive integer produces an expression called the DRF of If 
is the set of positive integers, 
is the set of DRF expressions, and the mapping defined by the factorization process is denoted 
then the doubly recursive factorization of is denoted 
The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called riffs and rotes, respectively.
 is the following digraph:
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 is the following graph:
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Riffs in Numerical Order
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Rotes in Numerical Order
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Prime Animations
Riffs 1 to 60
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Rotes 1 to 60
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Selected Sequences
A061396
- Number of "rooted index-functional forests" (Riffs) on n nodes.
- Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.
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![\begin{array}{lll}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\\[10pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}](/w/images/math/c/f/4/cf488a4a5059707198431623a68ff417.png)
![\begin{array}{lll}
\text{p}_4^1
& = & \text{p}_{\text{p}_1^2}^1
\\[10pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}](/w/images/math/a/d/6/ad6dea58d261c7fb0a769415bf3ebb13.png)
![\begin{array}{lll}
\text{p}_1^3
& = & \text{p}_1^{\text{p}_2^1}
\\[10pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}](/w/images/math/0/f/a/0fa95a9b9088dddb35a92f85329cd7d0.png)
![\begin{array}{lll}
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
\\[10pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}](/w/images/math/f/1/5/f150de2690bd260cfaa9b2858218d2df.png)
A062504
- Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.
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![\begin{array}{l|l|r}
k
& P_k
= \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \}
= \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \}
& |P_k|
\\[10pt]
0 & \{ 1 \} & 1
\\
1 & \{ 2 \} & 1
\\
2 & \{ 3, 4 \} & 2
\\
3 & \{ 5, 6, 7, 8, 9, 16 \} & 6
\\
4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20
\end{array}](/w/images/math/6/e/d/6ede82dcaa0dab7af2dd661cfacfdd8a.png)
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![\begin{array}{lll}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}](/w/images/math/7/9/f/79faa97e7c7d2664e275b5444f7084e4.png)
![\begin{array}{lll}
\text{p}_4^1
& = & \text{p}_{\text{p}_1^2}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}](/w/images/math/2/5/1/251943f1914b81df7a375496b170cdb8.png)
![\begin{array}{lll}
\text{p}_1^3
& = & \text{p}_1^{\text{p}_2^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}](/w/images/math/e/8/c/e8c741efc61e0f692fcf2ede19ff12ee.png)
![\begin{array}{lll}
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}](/w/images/math/7/b/d/7bd5094f9ef63f3b9db8994fc66a73f9.png)
![\begin{array}{lll}
\text{p}_1^1 \text{p}_3^1
& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1
\\[12pt]
& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}](/w/images/math/4/b/6/4b6fd63e12b9a4c4bd870421b8e2099b.png)
![\begin{array}{lll}
\text{p}_5^1
& = & \text{p}_{\text{p}_3^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1
\end{array}](/w/images/math/3/6/7/367f4bd3420c1bd0240ad8f364f078b9.png)
![\begin{array}{lll}
\text{p}_6^1
& = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1
\end{array}](/w/images/math/9/e/1/9e11ba3c6ad350ff590b47d89a748fb0.png)
![\begin{array}{lll}
\text{p}_1^1 \text{p}_4^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1
\\[12pt]
& = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}](/w/images/math/f/a/c/facdd8bf265818487986e48e628060dc.png)
![\begin{array}{lll}
\text{p}_7^1
& = & \text{p}_{\text{p}_4^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
\end{array}](/w/images/math/a/e/f/aef031c5d2f6e0b2c55f6b2418f33f83.png)
![\begin{array}{lll}
\text{p}_8^1
& = & \text{p}_{\text{p}_1^3}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1
\end{array}](/w/images/math/2/3/c/23c00d9e913d50c484db974f57502a53.png)
![\begin{array}{lll}
\text{p}_9^1
& = & \text{p}_{\text{p}_2^2}^1
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1
\end{array}](/w/images/math/5/0/6/5063f7a5994740dac288870c0f534d0d.png)
![\begin{array}{lll}
\text{p}_3^2
& = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1}
\\[12pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1}
\end{array}](/w/images/math/1/e/0/1e06bef980ff9e8f26527ea1104f70b5.png)
![\begin{array}{lll}
\text{p}_2^3
& = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1}
\\[12pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1}
\end{array}](/w/images/math/1/e/7/1e7ae62195eecf4c0e4b7274b1345513.png)
![\begin{array}{lll}
\text{p}_1^5
& = & \text{p}_1^{\text{p}_3^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}
\end{array}](/w/images/math/e/0/3/e03ac939a59ee30720fb4c7d612c5c3d.png)
![\begin{array}{lll}
\text{p}_4^2
& = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1}
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1}
\end{array}](/w/images/math/2/8/f/28f14521a894bfa9be1e05759b9f0bd7.png)
![\begin{array}{lll}
\text{p}_{16}^1
& = & \text{p}_{\text{p}_1^4}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1
\\[12pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1
\end{array}](/w/images/math/d/e/1/de1e44673d70fe2aeb90639ff85e45ad.png)
![\begin{array}{lll}
\text{p}_1^6
& = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
\end{array}](/w/images/math/7/b/9/7b93cc28331ae3389c2055633c1cdb28.png)
![\begin{array}{lll}
\text{p}_2^4
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2}
\\[12pt]
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}}
\end{array}](/w/images/math/5/d/8/5d86ed3cd59d4f68ec1f6cb457aa6cff.png)
![\begin{array}{lll}
\text{p}_1^7
& = & \text{p}_1^{\text{p}_4^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}
\end{array}](/w/images/math/6/9/f/69f95810ac992edb0448c872daae91e2.png)
![\begin{array}{lll}
\text{p}_1^8
& = & \text{p}_1^{\text{p}_1^3}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}
\end{array}](/w/images/math/3/c/4/3c47d970a0affa24a768580b65a4261a.png)
![\begin{array}{lll}
\text{p}_1^9
& = & \text{p}_1^{\text{p}_2^2}
\\[12pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}
\end{array}](/w/images/math/f/b/7/fb78d8a35e91af8e68924bedfef8f43f.png)
![\begin{array}{lll}
\text{p}_1^{16}
& = & \text{p}_1^{\text{p}_1^4}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}}
\\[12pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}
\end{array}](/w/images/math/b/e/c/bec8d073c0eff26826a7597a632c73c2.png)
A062537
- Nodes in riff (rooted index-functional forest) for n.
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A062860
- Smallest j with n nodes in its riff (rooted index-functional forest).
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A109301
- 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.
- Example
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Miscellaneous Examples
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