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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 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.)