Riffs & Rotes & A061396 & A062504?
Jon Awbrey
jawbrey at oakland.edu
Mon Jun 25 07:48:33 CEST 2001
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Neil,
The latest tables are appended below.
This data allows us to extend the inverse table another row:
o--------------------------------------------------------------------------------
Notes on A062504 [pending]
Triangle in which k-th row lists natural number
values for the collection of riffs with k nodes:
k | natural numbers n such that |riff(n)| = k
--o------------------------------------------------------------------------------
0 | 1;
1 | 2;
2 | 3, 4;
3 | 5, 6, 7, 8, 9, 16;
4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25,
| 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536;
5 | 15, 20, 21, 22, 24, 26, 28, 29, 31, 34,
| 36, 37, 38, 41, 43, 46, 48, 50, 54, 59,
| 61, 67, 83, 97, 98, 103, 106, 121, 125, 131,
| 162, 169, 227, 241, 243, 289, 311, 343, 361, 419,
| 529, 625, 719, 729, 1024, 1619, 2048, 2187, 2401, 2809,
| 3671, 4096, 6561, 8192, 16384, 19683, 131072, 262144, 524288, 821641,
| 8388608, 33554432, 43046721, 134217728, 4294967296, 562949953421312,
| 9007199254740992, 18446744073709551616, 2417851639229258349412352,
| 2^128, 2^256, 2^512, 2^65536;
|
--o------------------------------------------------------------------------------
The natural number values for the riffs with
at most 3 pts are as follows (@'s are roots):
| o o o o
| | ^ | ^
| v | v |
| o o o o o o o o o
| | ^ | | | ^ | ^ ^
| v | v v v | v/ |
| Riff: @; @, @; @, @ @, @, @, @, @;
|
| Value: 2; 3, 4; 5, 6 , 7, 8, 9, 16;
o--------------------------------------------------------------------------------
Minimum and Maximum Values on Row k?
Min: 1, 2, 3, 5, 10, 15, ...
Need more terms to distinguish this
from a welter of other sequences.
Max: 1, 2, 4, 16, 65536, 2^65536, ...
Unless there is some sort of number-theoretic surprise
that I cannot foresee, the maximum sequence should be
A014221: a(n) = 2^a(n-1).
o--------------------------------------------------------------------------------
Tables for Reference
Table 1. Riffs & Rotes Up To Three Nodes
o--------------------------------------------------------------------------------
| integer factorization riff r.i.f.f. rote --> in parentheses
| k p's k nodes 2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1 1 blank blank @ blank
|
o--------------------------------------------------------------------------------
|
| o---o
| |
| 2 p_1^1 p @ @ (())
|
o--------------------------------------------------------------------------------
|
| o---o
| |
| o---o
| 3 p_2^1 = |
| p_(p_1)^1 p< @ @ ((())())
| p ^
| \
| o
|
| o---o
| o |
| ^ o---o
| 4 p_1^2 = p / |
| p_1^p_1 p< @ @ (((())))
|
o--------------------------------------------------------------------------------
|
| o---o
| |
| o---o
| |
| 5 p_3 = o---o
| p_(p_2) = |
| p_(p_(p_1)) p< @ @ (((())())())
| p< ^
| p \
| o
| ^
| \
| o
|
| o-o
| /
| o-o o-o
| 6 p_1 p_2 = \ /
| p_1 p_(p_1) p.p< @ @ @ (())((())())
| p ^
| \
| o
|
| o---o
| |
| o---o
| |
| 7 p_4 = o---o
| p_(p_1^2) = |
| p_(p_1^p_1) p< p @ o @ ((((())))())
| p< ^ ^
| \ /
| o
|
| o---o
| |
| o---o
| o |
| 8 p_1^3 = ^ ^ o---o
| p_1^p_2 = p< / \ |
| p_1^p_(p_1) p< p @ o @ ((((())())))
|
| o-o o-o
| o | |
| 9 p_2^2 = ^ o---o
| p_(p_1)^2 = p / |
| p_(p_1)^(p_1) p< @ @ ((())((())))
| p ^
| \
| o
|
| o o---o
| ^ |
| / o---o
| o |
| 16 p_1^4 = p ^ o---o
| p_1^(p_1^2) = p< / |
| p_1^(p_1^p_1) p< @ @ (((((())))))
|
o--------------------------------------------------------------------------------
Table 2. Riffs Up To Five Nodes
o================================================================================
| k = 0
| blank
|
| 1
| 1
o================================================================================
| k = 1
| p
|
| 2
| 1
o================================================================================
| k = 2
| p
| p< p<
| p
|
| 2^2 p_2
| 4 3
| 2
o================================================================================
| k = 3
| p
| p< p<
| p< p< p p< p p<
| p< p<
| p
|
| 2^4 2^3 p_4 p_3
| 16 8 7 5
| 4
o--------------------------------------------------------------------------------
|
| p
| p< p.p<
| p p
|
| p_2^2 2 p_2
| 9 6
| 6
o================================================================================
| k = 4
| p
| p< p p<
| p< p< p< p p< p p.p< p<
| p< p< p p< p< p< p< p p< p<
| p
|
| 2^16 2^9 2^8 2^7 2^6 2^5
| 65536 512 256 128 64 32
| 6
o--------------------------------------------------------------------------------
|
| p
| p< p< p< p p< p< p< p< p<
| p< p< p< p p< p p.p< p<
| p p< p p<
| p
|
| p_16 p_9 p_8 p_7 p_6 p_5
| 53 23 19 17 13 11
| 12
o--------------------------------------------------------------------------------
|
| p
| p< p< p p
| p< p< p p< p p<
| p p p< p<
| p
|
| 3^4 3^3 7^2 5^2
| 81 27 49 25
| 16
o--------------------------------------------------------------------------------
|
| p p
| p.p< p.p< p p.p< p< .p<
| p p< p< p
| p
|
| 2 3^2 2 p_4 2 p_3 2^2 p_2
| 18 14 10 12
| 20
o================================================================================
| k = 5
| p
| p< p p<
| p< p< p< p p< p p.p< p<
| p< p< p p< p< p< p< p p< p<
| p< p< p< p< p< p< p
|
| 2^65536 2^512 2^256 2^128 2^64 2^32
| 6
o--------------------------------------------------------------------------------
|
| p
| p< p< p< p p< p< p< p< p<
| p< p< p< p< p< p< p p< p< p p< p.p< p< p<
| p p< p p<
| p
|
| 2^53 2^23 2^19 2^17 2^13 2^11
| 12
o--------------------------------------------------------------------------------
|
| p
| p< p< p p
| p< p< p p< p p<
| p< p p< p p< p< p< p<
| p
|
| 2^81 2^27 2^49 2^25
| 16
o--------------------------------------------------------------------------------
|
| p p
| p.p< p.p< p p.p< p< .p<
| p< p p< p< p< p< p< p
| p
|
| 2^18 2^14 2^10 2^12
| 20
o--------------------------------------------------------------------------------
|
| p
| p< p p<
| p< p< p< p< p< p< p p< p< p p< p.p< p< p<
| p< p< p p< p< p< p< p p< p<
| p
|
| p_65536 p_512 p_256 p_128 p_64 p_32
| 821641 3671 1619 719 311 131
| 26
o--------------------------------------------------------------------------------
|
| p< p p< p< p< p< p<
| p< p< p< p p< p< p< p< p<
| p< p< p< p p< p p p< p<
| p p< p p<
| p
|
| p_53 p_23 p_19 p_17 p_13 p_11
| 241 83 67 59 41 31
| 32
o--------------------------------------------------------------------------------
|
| p
| p< p< p< p< p< p p< p
| p< p< p p< p p<
| p p p< p<
| p
|
| p_81 p_27 p_49 p_25
| 419 103 227 97
| 36
o--------------------------------------------------------------------------------
|
| p< p p< p< p< p
| p.p< p.p< p p.p< p< .p<
| p p< p< p
| p
|
| p_18 p_14 p_10 p_12
| 61 43 29 37
| 40
o--------------------------------------------------------------------------------
|
| p
| p< p p<
| p< p< p< p p< p p.p< p<
| p< p< p p< p< p< p< p p< p<
| p p p p p p p
|
| 3^16 3^9 3^8 3^7 3^6 3^5
| 43046721 19683 6561 2187 729 243
| 46
o--------------------------------------------------------------------------------
|
| p p p p p p p
| p< p< p< p p< p< p< p< p<
| p< p< p< p p< p p.p< p<
| p p< p p<
| p
|
| p_16^2 p_9^2 p_8^2 p_7^2 p_6^2 p_5^2
| 2809 529 361 289 169 121
| 52
o--------------------------------------------------------------------------------
|
| p
| p.p< p< p.p< p p.p< p< p.p< p.p< p.p<
| p< p< p< p p< p p.p< p<
| p p< p p<
| p
|
| 2 p_16 2 p_9 2 p_8 2 p_7 2 p_6 2 p_5
| 106 46 38 34 26 22
| 58
o--------------------------------------------------------------------------------
|
| p
| p< p< p p
| p.p< p.p< p p.p< p p.p<
| p p p< p<
| p
|
| 2 3^4 2 3^3 2 p_4^2 2 p_3^2
| 162 54 98 50
| 62
o--------------------------------------------------------------------------------
|
| p
| p< p< p p
| p< .p< p< p .p< p< .p< p p< .p<
| p p p< p<
| p
|
| 2^4 3 2^3 3 4 p_4 4 p_3
| 48 24 28 20
| 66
o--------------------------------------------------------------------------------
|
| p^p p_p p^p p_p
| p< p< p< p<
| p^p p^p p_p p_p
|
| p_4^4 p_4^3 p_3^4 p_3^3
| 2401 343 625 125
| 70
o--------------------------------------------------------------------------------
|
| p p
| p< .p< p p< .p< p< .p<
| p p< p p< p
| p
|
| 3 p_4 3 p_3 2^2 3^2
| 21 15 36
| 73
o================================================================================
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