A323608 The position function the fractalization of which yields A323607.

1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 8, 6, 9, 7, 12, 8, 12, 9, 15, 10, 15, 11, 19, 12, 18, 13, 22, 14, 21, 15, 27, 16, 24, 17, 29, 18, 27, 19, 34, 20, 30, 21, 36, 22, 33, 23, 42, 24, 36, 25, 43, 26, 39, 27, 49, 28, 42, 29, 50, 30, 45, 31, 58, 32, 48, 33, 57, 34, 51, 35, 64, 36, 54, 37, 64, 38, 57, 39, 73

Offset

1,4

Comments

For a definition of the fractalization process, see comments in A194959. The sequence A323607, triangular array where row n is the list of the numbers from 1 to n sorted in Sharkovsky order, is clearly the result of a fractalization. Let {a(n)} (this sequence) be its position function.

Links

Table of n, a(n) for n=1..80.

Formula

Empirical observations: (Start)

For all odd numbers x >= 3,

a(x) = (1/2)*x - 1/2,

a(2x) = (3/4)*(2x) - 3/2,

a(4x) = (7/8)*(4x) - 5/2,

a(8x) = (15/16)*(8x) - 7/2,

etc.

For all c, a(2^c) = A000325(c) = 2^c-c.

Summarized by:

a((2^c)*(2k+1)) = A126646(c)*k + A000295(c) + A000007(k) = (2^(c+1)-1)*k + (2^c-1-c) + [k==0].

(End)

From Luc Rousseau, Apr 01 2019: (Start)

It appears that for all k > 0,

a(4k + 0) = 3k - 2 + a(k),

a(4k + 1) = 2k,

a(4k + 2) = 3k,

a(4k + 3) = 2k + 1.

(End)

Example

In A323607 in triangular form,
- row 5 is:  3  5  4  2  1
- row 6 is:  3  5  6  4  2  1
Row 6 is row 5 in which 6 has been inserted in position 3, so a(6) = 3.

Mathematica

lt[x_, y_] := Module[
  {c, d, xx, yy, u, v},
  {c, d} = IntegerExponent[#, 2] & /@ {x, y};
  xx = x/2^c;
  yy = y/2^d;
  u = If[xx == 1, \[Infinity], c];
  v = If[yy == 1, \[Infinity], d];
  If[u != v, u < v, If[u == \[Infinity], c > d, xx < yy]]]
row[n_] := Sort[Range[n], lt]
a[n_] := First[FirstPosition[row[n], n]]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A194959 (introducing fractalization).

Cf. A323607 (fractalization of this sequence).

Cf. A126646, A000295, A000007.

Cf. A000325.

Sequence in context: A213634 A300271 A252461 * A122352 A338903 A362830

Adjacent sequences: A323605 A323606 A323607 * A323609 A323610 A323611

Keyword

nonn,look

Author

Luc Rousseau, Jan 19 2019

Status

approved