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 A038568 Numerators in canonical bijection from positive integers to positive rationals. 11
 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 4, 1, 5, 2, 5, 3, 5, 4, 5, 1, 6, 5, 6, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 1, 8, 3, 8, 5, 8, 7, 8, 1, 9, 2, 9, 4, 9, 5, 9, 7, 9, 8, 9, 1, 10, 3, 10, 7, 10, 9, 10, 1, 11, 2, 11, 3, 11, 4, 11, 5, 11, 6, 11, 7, 11, 8, 11, 9, 11, 10, 11, 1, 12, 5, 12, 7, 12, 11, 12, 1, 13, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Even-indexed terms are positive integers in order, with m occurring phi(m) times. Preceding odd-indexed terms (except for missing initial 0) are the corresponding numbers <= m and relatively prime to m, in increasing order. The denominators are just this sequence shifted left. Thus each positive rational occurs exactly once as a ratio a(n)/a(n+1). - Franklin T. Adams-Watters, Dec 06 2006 REFERENCES H. Lauwerier, Fractals, Princeton Univ. Press, p. 23. LINKS David Wasserman, Table of n, a(n) for n = 0..100000 EXAMPLE First arrange fractions by increasing denominator, then by increasing numerator: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ... (this is A038566/A038567); now follow each term (except the first) with its reciprocal: 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, ... (this is A038568/A038569). MAPLE with (numtheory): A038568 := proc (n) local sum, j, k; sum := 1: k := 2: while (sum < n) do: sum := sum + 2 * phi(k): k := k + 1: od: sum := sum - 2 * phi(k-1): j := 1: while sum < n do: if gcd(j, k-1) = 1 then sum := sum + 2: fi: j := j+1: od: if sum > n then RETURN (j-1) fi: RETURN (k-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com) MATHEMATICA a[n_] := Module[{sum = 1, k = 2}, While[sum < n, sum = sum + 2*EulerPhi[k]; k = k+1]; sum = sum - 2*EulerPhi[k-1]; j = 1; While[sum < n, If[GCD[j, k-1] == 1, sum = sum+2]; j = j+1; ]; If[sum > n, Return[j-1]]; Return[k-1] ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 21 2012, translated from Maple *) PROG (Python) from sympy import totient, gcd def a(n):     s=1     k=2     while sn: return j - 1     return k - 1 # Indranil Ghosh, May 23 2017, translated from Mathematica (Julia) using Nemo function A038568List(len)     a, A = QQ(0), []     for n in 1:len         a = next_minimal(a)         push!(A, numerator(a))     end A end A038568List(84) |> println # Peter Luschny, Mar 13 2018 (PARI) a(n) = { my (e); for (q=1, oo, if (n+1<2*e=eulerphi(q), for (p=1, oo, if (gcd(p, q)==1, if (n+1<2, return ([p, q][n+2]), n-=2))), n-=2*e)) } \\ Rémy Sigrist, Feb 25 2021 CROSSREFS Cf. A020652, A020653, A038566, A038567, A038569. Sequence in context: A215467 A284266 A317988 * A071912 A070940 A020651 Adjacent sequences:  A038565 A038566 A038567 * A038569 A038570 A038571 KEYWORD nonn,frac,core,easy,nice AUTHOR EXTENSIONS More terms from Erich Friedman STATUS approved

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Last modified October 6 15:48 EDT 2022. Contains 357269 sequences. (Running on oeis4.)