Shifts and signs: Difference between revisions
New page: The first 1124 sequence seems to exhibit great structure in its subsequences of the form <math>x_{p^k m}</math>. If <math>m \nmid p</math>, we can often (subject to some anomalies) write ... |
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where <math>\eta(p,m) = \pm 1</math> and <math>s(p,m) \geq 0</math>. | where <math>\eta(p,m) = \pm 1</math> and <math>s(p,m) \geq 0</math>. | ||
This table shows the first few values of <math>\eta(p,m)</math> and <math>s(p,m)</math> | This table shows the first few values of <math>\eta(p,m)</math> and <math>s(p,m)</math>. Sometimes one has a choice; this table reflects a preference for multiplicativity of <math>\eta(p,m)</math> as a function of <math>m</math> and monotictity of <math>s(p,m)</math> as a function of <math>m</math>. | ||
<pre> | <pre> | ||
| Line 12: | Line 12: | ||
2 1 -1 2 +1 1 +1 | 2 1 -1 2 +1 1 +1 | ||
3 1 -1 2 -1 1 -1 | 3 1 -1 2 -1 1 -1 | ||
4 2 +1 1 | 4 2 +1 3 +1 2 +1 | ||
5 2 -1 2 -1 | 5 2 -1 2 -1 2 -1 | ||
6 | 6 3 -1 2 -1 | ||
7 3 -1 ? | 7 3? -1? 4 +1 2/4 +1/-1 | ||
8 3 -1 3 -1 | 8 3 -1 3 -1 3 +1 | ||
9 2 +1 1 | 9 2 +1 3 +1 2 +1 | ||
10 3 +1 | 10 3 +1 | ||
11 3 +1 3 -1 | 11 3 +1 3 -1 | ||
| Line 33: | Line 33: | ||
23 3 +1 | 23 3 +1 | ||
24 | 24 | ||
25 4 | 25 4 +1 | ||
26 | 26 | ||
27 3 -1 | 27 3 -1 | ||
| Line 41: | Line 41: | ||
31 5 -1 | 31 5 -1 | ||
32 | 32 | ||
33 4 | 33 4? -1? | ||
34 | 34 | ||
35 5 +1 | 35 5 +1 | ||
Latest revision as of 10:07, 17 January 2010
The first 1124 sequence seems to exhibit great structure in its subsequences of the form [math]\displaystyle{ x_{p^k m} }[/math]. If [math]\displaystyle{ m \nmid p }[/math], we can often (subject to some anomalies) write
[math]\displaystyle{ x_{p^k m} = \eta(p,m) x_{p^{k+s(p,m)}} }[/math]
where [math]\displaystyle{ \eta(p,m) = \pm 1 }[/math] and [math]\displaystyle{ s(p,m) \geq 0 }[/math].
This table shows the first few values of [math]\displaystyle{ \eta(p,m) }[/math] and [math]\displaystyle{ s(p,m) }[/math]. Sometimes one has a choice; this table reflects a preference for multiplicativity of [math]\displaystyle{ \eta(p,m) }[/math] as a function of [math]\displaystyle{ m }[/math] and monotictity of [math]\displaystyle{ s(p,m) }[/math] as a function of [math]\displaystyle{ m }[/math].
m s(2,m) e(2,m) s(3,m) e(3,m) s(5,m) e(5,m) s(7,m) e(7,m) 1 0 +1 0 +1 0 +1 0 +1 2 1 -1 2 +1 1 +1 3 1 -1 2 -1 1 -1 4 2 +1 3 +1 2 +1 5 2 -1 2 -1 2 -1 6 3 -1 2 -1 7 3? -1? 4 +1 2/4 +1/-1 8 3 -1 3 -1 3 +1 9 2 +1 3 +1 2 +1 10 3 +1 11 3 +1 3 -1 12 13 3 -1 3 +1 14 4 -1 15 3 +1 16 4 +1 17 4 -1 4 -1 18 19 3 -1 20 21 4 +1 22 23 3 +1 24 25 4 +1 26 27 3 -1 28 29 5 +1 30 31 5 -1 32 33 4? -1? 34 35 5 +1 36 37 4 +1 38 39 4 +1 40 41 5 +1
