Shifts and signs

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	0/2	+1
5	2	-1	2	-1			0/2	-1
6					3	-1	0/2	-1
7	3?	-1?	0	-1	0	-1
8			3	-1	3	-1	1/3	+1
9	2	+1			3	+1	0/2	+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