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A323608 The position function the fractalization of which yields A323607. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A248519

Adjacent sequences:  A323605 A323606 A323607 * A323609 A323610 A323611

KEYWORD

nonn,look

AUTHOR

Luc Rousseau, Jan 19 2019

STATUS

approved

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Last modified August 1 03:52 EDT 2021. Contains 346384 sequences. (Running on oeis4.)