

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
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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^cc.
Summarized by:
a((2^c)*(2k+1)) = A126646(c)*k + A000295(c) + A000007(k) = (2^(c+1)1)*k + (2^c1c) + [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



