A191774 Lim f(f(...f(n)...)) where f(n) is the Farey fractal sequence, A131967.

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, 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, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..86.

Wikipedia, Fractal sequence

Example

Write the counting numbers and A131967 like this:
1..2..3..4..5..6..7..8..9..10..11..12..13..14..15..
1..2..1..3..2..1..4..3..5..2...1...6...4...3...5...
It is then easy to check composites:
1->1, 2->2, 3->1, 4->3->1, 5->2, 6->1, 7->4->3->1,...

Mathematica

Farey[n_] := Select[Union@Flatten@Outer[Divide, Range[n + 1] - 1, Range[n]], # <= 1 &];
newpos[n_] := Module[{length = Total@Array[EulerPhi, n] + 1, f1 = Farey[n], f2 = Farey[n - 1], to},
   to = Complement[Range[length], Flatten[Position[f1, #] & /@ f2]];
   ReplacePart[Array[0 &, length],
    Inner[Rule, to, Range[length - Length[to] + 1, length], List]]];
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 = a[12]; f[n_] := Part[t, n];
Table[f[n], {n, 1, 100}]          (* A131967 *)
h[n_] := Nest[f, n, 50]
t = Table[h[n], {n, 1, 200}]      (* A191774 *)
s = Flatten[Position[t, 1]]       (* A191775 *)
s = Flatten[Position[t, 2]]       (* A191776 *)

Crossrefs

Cf. A020903, A191770, A191775, A191776.

Sequence in context: A020906 A220280 A355242 * A262885 A097305 A120675

Adjacent sequences: A191771 A191772 A191773 * A191775 A191776 A191777

Keyword

nonn

Author

Clark Kimberling, Jun 16 2011

Status

approved