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A177903 Consider the weighted Farey tree A177405/A177407; a(n) = row at which the denominator 2n+1 first appears (assumes first row is labeled row 0).

%I A177903 #27 Sep 21 2017 09:03:23
%S A177903 0,1,2,2,2,3,3,4,3,3,4,4,4,3,4,4,5,5,5,5,4,4,5,4,5,6,4,4,6,5,6,5,5,5,
%T A177903 5,5,5,5,5,5,4,5,6,7,6,6,6,6,6,5,6,5,6,6,6,6,5,6,7,6,6,6,6,6,6,6,5,6,
%U A177903 5,7,7,6,6,7,7,6,7,6,6,6,5,5,7,6,6,6,7,7,7,6,6,6,7,7,6,7,7,7,6,7,7
%N A177903 Consider the weighted Farey tree A177405/A177407; a(n) = row at which the denominator 2n+1 first appears (assumes first row is labeled row 0).
%C A177903 Latest occurrences of odd denominators 1,3,5,7,...,29: 0,1,3,3,4,5,6,7,8,9,10,11,12,13,14,15 (The glitch in the third term reflects the fact that 2/5 and 3/5 don't show up until the 3rd iteration; whereas for n>2, it appears that the last fraction with denominator 2n+1 to show up is 1/(2n+1), and that this fraction shows up after exactly n iterations.) - James Propp
%D A177903 Based on postings by Richard C. Schroeppel and James Propp to the Math Fun Mailing List, Dec 15 2010.
%t A177903 Denom[L_, k_] :=
%t A177903 Module[{M, i}, M = {};
%t A177903   For[i = 1, i <= Length[L], i++,
%t A177903    If[Denominator[L[[i]]] == k, M = Append[M, L[[i]]]]]; Return[M]]
%t A177903 Earliest[k_] :=
%t A177903 Module[{i}, For[i = 1, Length[Denom[WF[i], k]] == 0, i++]; Return[i]]
%t A177903 Latest[k_] :=
%t A177903 Module[{i}, For[i = 1, Length[Denom[WF[i], k]] < EulerPhi[k], i++];
%t A177903   Return[i]]
%t A177903 Table[Earliest[2 n + 1], {n, 1, 100}]
%t A177903 (* _James Propp_ *)
%Y A177903 Cf. A177405, A177407. See A178042 for another version. Cf. also A178031.
%K A177903 nonn
%O A177903 0,3
%A A177903 _N. J. A. Sloane_, Dec 15 2010