# Shifts and signs

From Polymath1Wiki

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

[math]x_{p^k m} = \eta(p,m) x_{p^{k+s(p,m)}}[/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]. 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].

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