A323607 Triangular array: row n is the list of numbers from 1 to n, sorted in Sharkovsky order.

1, 2, 1, 3, 2, 1, 3, 4, 2, 1, 3, 5, 4, 2, 1, 3, 5, 6, 4, 2, 1, 3, 5, 7, 6, 4, 2, 1, 3, 5, 7, 6, 8, 4, 2, 1, 3, 5, 7, 9, 6, 8, 4, 2, 1, 3, 5, 7, 9, 6, 10, 8, 4, 2, 1, 3, 5, 7, 9, 11, 6, 10, 8, 4, 2, 1, 3, 5, 7, 9, 11, 6, 10, 12, 8, 4, 2, 1, 3, 5, 7, 9, 11, 13, 6, 10, 12, 8, 4, 2, 1

Offset: 1

Author

Luc Rousseau, Jan 19 2019

Keywords

• nonn
• tabl

Examples

			Array begins:
  1
  2  1
  3  2  1
  3  4  2  1
  3  5  4  2  1
  3  5  6  4  2  1
  3  5  7  6  4  2  1
  3  5  7  6  8  4  2  1
  3  5  7  9  6  8  4  2  1
  3  5  7  9  6 10  8  4  2  1
  3  5  7  9 11  6 10  8  4  2  1
  3  5  7  9 11  6 10 12  8  4  2  1
		

Links

• Luc Rousseau, Array plot of the first 250 rows of the triangle
• Eric Weisstein's World of Mathematics, Sharkovsky's Theorem

Crossrefs

Cf. A323608.

Programs

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]
row /@ Range[13] // Flatten