%I A038568
%S A038568 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,
%T A038568 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,
%U A038568 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
%N A038568 Numerators in canonical bijection from positive integers to positive 
               rationals.
%C A038568 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 (FrankTAW(AT)Netscape.net), 
               Dec 06 2006
%D A038568 H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.
%H A038568 David Wasserman, <a href="b038568.txt">Table of n, a(n) for n = 0..100000</
               a>
%H A038568 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%e A038568 First arrange fractions by increasing denominator then by increasing 
               numerator:
%e A038568 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ... (this is A038566/A038567);
%e A038568 now follow each term by its reciprocal:
%e A038568 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).
%p A038568 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: # from UlrSchimke(AT)aol.com, Oct 31, 2001
%Y A038568 Cf. A020652, A020653, A038566-A038569.
%Y A038568 Sequence in context: A057940 A097285 A057432 this_sequence A071912 A070940 
               A020651
%Y A038568 Adjacent sequences: A038565 A038566 A038567 this_sequence A038569 A038570 
               A038571
%K A038568 nonn,frac,core,easy,nice
%O A038568 0,3
%A A038568 N. J. A. Sloane (njas(AT)research.att.com).
%E A038568 More terms from Erich Friedman (erich.friedman(AT)stetson.edu).