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A191770 Lim f(f(...f(n)...)) where f(n) is the fractal sequence A022446. 6
1, 2, 3, 1, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 3, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 1, 3, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 3, 2, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Suppose that f(1), f(2), ... is a fractal sequence (such as 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ..., which contains itself as a proper subsequence - 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}.  If f(2)>2, then L(n)=1 for all n; if f(2)=2 and f(3)>3, then L(n) equals 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 A022446 like this:

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

1..2..3..1..4..2..5..8..1..4...6...2...7...5...3...

It is then easy to check composites:

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

MATHEMATICA

g[n_] :=  Length[Select[Table[FixedPoint[i + PrimePi[#] + 1 &, i + PrimePi[i] + 1], {i, n}], # <= n &]];

f[n_] := PrimePi[NestWhile[g, n, ! PrimeQ[#] && # != 1 &]] + 1;

Array[f, 80]             (* A022446 *)

h[n_] := Nest[f, n, 40]; t = Table[h[n], {n, 1, 300}]  (* A191770 *)

Flatten[Position[t, 1]]  (* A191771 *)

Flatten[Position[t, 2]]  (* A191772 *)

Flatten[Position[t, 3]]  (* A191773 *)

CROSSREFS

Cf. A020903, A191770, A191771, A191772, A191773, A191774.

Sequence in context: A113925 A180466 A105083 * A120966 A189041 A055444

Adjacent sequences:  A191767 A191768 A191769 * A191771 A191772 A191773

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 16 2011

STATUS

approved

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Last modified April 24 00:02 EDT 2019. Contains 322404 sequences. (Running on oeis4.)