A191774 - OEIS

%I #5 Mar 30 2012 18:57:33

%S 1,2,1,1,2,1,1,1,2,2,1,1,1,1,2,1,2,1,1,1,1,2,1,2,2,1,1,2,1,1,1,1,1,2,

%T 1,2,2,1,1,1,2,1,1,1,1,1,2,1,2,1,1,2,2,2,1,1,1,1,1,2,1,1,1,1,1,1,2,1,

%U 1,2,1,1,2,2,2,2,1,1,1,1,1,1,2,1,2,1

%N Lim f(f(...f(n)...)) where f(n) is the Farey fractal sequence, A131967.

%C Suppose that f(1), f(2), f(3),... is a fractal sequence (a sequence which contains itself as a proper subsequence, such as 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ...; if the first occurrence of each n is deleted, the remaining sequence is identical to the original; see the Wikipedia article for a rigorous definition).  Then for each n>=1, the limit L(n) of composites f(f(f...f(n)...)) exists and is one of the numbers in the set {k : f(k)=k}.  Thus, if f(2)>2, then L(n)=1 for all n; if f(2)=2 and f(3)>3, then L(n) is 1 or 2 for all n.  Examples:  A020903, A191770, A191774

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fractal_sequence">Fractal sequence</a>

%e Write the counting numbers and A131967 like this:

%e 1..2..3..4..5..6..7..8..9..10..11..12..13..14..15..

%e 1..2..1..3..2..1..4..3..5..2...1...6...4...3...5...

%e It is then easy to check composites:

%e 1->1, 2->2, 3->1, 4->3->1, 5->2, 6->1, 7->4->3->1,...

%t Farey[n_] := Select[Union@Flatten@Outer[Divide, Range[n + 1] - 1, Range[n]], # <= 1 &];

%t newpos[n_] := Module[{length = Total@Array[EulerPhi, n] + 1, f1 = Farey[n], f2 = Farey[n - 1], to},

%t    to = Complement[Range[length], Flatten[Position[f1, #] & /@ f2]];

%t    ReplacePart[Array[0 &, length],

%t     Inner[Rule, to, Range[length - Length[to] + 1, length], List]]];

%t a[n_] := Flatten@Table[Fold[ReplacePart[Array[newpos, i][[#2 + 1]], Inner[Rule, Flatten@Position[Array[newpos, i][[#2 + 1]], 0], #1, List]] &, Array[newpos, i][[1]], Range[i - 1]], {i, n}];

%t t = a[12]; f[n_] := Part[t, n];

%t Table[f[n], {n, 1, 100}]          (* A131967 *)

%t h[n_] := Nest[f, n, 50]

%t t = Table[h[n], {n, 1, 200}]      (* A191774 *)

%t s = Flatten[Position[t, 1]]       (* A191775 *)

%t s = Flatten[Position[t, 2]]       (* A191776 *)

%Y Cf. A020903, A191770, A191775, A191776.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jun 16 2011