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A226137
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Positions of the integers in the ordering of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)
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4
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1, 2, 3, 4, 6, 7, 10, 14, 15, 22, 32, 46, 47, 69, 101, 147, 148, 217, 318, 465, 466, 683, 1001, 1466, 1467, 2150, 3151, 4617, 4618, 6768, 9919, 14536, 14537, 21305, 31224, 45760, 45761, 67066, 98290, 144050, 144051, 211117, 309407, 453457, 453458
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OFFSET
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1,2
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COMMENTS
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Let S be the set of numbers defined by these rules: 1 is in S, and if nonzero x is in S, then x + 1 and -1/x are in S. Then S is the set of all rational numbers, produced in generations as follows: g(1) = (1), g(2) = (2, -1), g(3) = (3, -1/2, 0), g(4) = (4, -1/3, 1/2), ... For n > 4, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements. Let S' denote the sequence formed by concatenating the generations.
A226130: Denominators of terms of S'
A226136: Positions of positive integers in S'
A226137: Positions of integers in S'
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LINKS
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EXAMPLE
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S'= (1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...), with integers appearing in positions 1,2,3,4,6,7,...
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MATHEMATICA
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g[1] := {1}; z = 20; g[n_] := g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]]&[DeleteCases[g[n - 1], 0]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; Flatten[Map[g, Range[7]]] (* ordered rationals *)
Map[g, Range[z]]; Table[Length[g[i]], {i, 1, z}] (* cf A003410 *)
f = Flatten[Map[g, Range[z]]];
Take[Denominator[f], 100] (* A226130 *)
Take[Numerator[f], 100] (* A226131 *)
p1 = Flatten[Table[Position[f, n], {n, 1, z}]] (* A226136 *)
p2 = Flatten[Table[Position[f, -n], {n, 0, z}]];
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CROSSREFS
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Cf. A226080 (rabbit ordering of positive rationals).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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