A131967 -id:A131967 - cp's OEIS frontend

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

Clark Kimberling, Jun 16 2011

nonn

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

			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,...

Wikipedia, Fractal sequence

Cf. A020903, A191770, A191775, A191776.

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 *)

A131968 Interspersion of the Farey fractal sequence, A131967.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 11, 10, 8, 7, 18, 17, 14, 13, 9, 29, 28, 23, 21, 15, 12, 42, 41, 35, 33, 25, 20, 16, 61, 60, 51, 48, 37, 32, 26, 19, 86, 85, 71, 68, 54, 46, 38, 31, 22

Offset: 1

Clark Kimberling, Aug 02 2007

A permutation of the natural numbers.

			Northwest corner:
1 3 6 11 18
2 5 10 17 28
4 8 14 23 35
7 13 21 33 48

C. Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

Cf. A131967.

Rectangular array T given by antidiagonals. T(i,j) = the j-th index n such that A131967(n)=i.

A049688 a(n) = Sum_{i=0..n} T(i,n-i), array T as in A049687.

Original entry on oeis.org

0, 2, 5, 10, 17, 28, 41, 60, 83, 112, 145, 188, 235, 294, 359, 432, 513, 610, 713, 834, 963, 1104, 1255, 1428, 1609, 1810, 2023, 2254, 2497, 2768, 3047, 3356, 3681, 4026, 4387, 4772, 5169, 5602, 6053, 6528, 7019

Offset: 0

Clark Kimberling

nonn

A131967(a(n)+1) = 1, A131967(a(n)) = 2. - Birkas Gyorgy, Feb 19 2011

Number of triples {A, B, C} where 1 <= A <= B <= C <= n+1 and gcd(C-B, B-A) = 1. E.g., for n=2, we have the 5 triples {1, 1, 2}, {1, 2, 2}, {2, 2, 3}, {2, 3, 3}, and {1, 2, 3}. - Griffin N. Macris, May 21 2016

Griffin N. Macris, Table of n, a(n) for n = 0..99999

Cf. A049687, A131967.

Table[Sum[Sum[EulerPhi[j], {j, i}] + 1, {i, n}], {n, 0, 30}] (* Birkas Gyorgy, Feb 19 2011 *)
Table[n + Sum[ EulerPhi[ j], {i, n}, {j, i}], {n, 0, 30}] (* Robert G. Wilson v, Feb 12 2015 *)

a(n) ~ n^2*(3+n) / Pi^2. - Griffin N. Macris, May 21 2016

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

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A131968 Interspersion of the Farey fractal sequence, A131967.

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A049688 a(n) = Sum_{i=0..n} T(i,n-i), array T as in A049687.

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