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).

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, 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, 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

Offset: 0

Author

N. J. A. Sloane, Dec 15 2010

Keywords

• nonn

Comments

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

References

• Based on postings by Richard C. Schroeppel and James Propp to the Math Fun Mailing List, Dec 15 2010.

Crossrefs

Cf. A177405, A177407. See A178042 for another version. Cf. also A178031.

Programs

Mathematica

Denom[L_, k_] :=
Module[{M, i}, M = {};
  For[i = 1, i <= Length[L], i++,
   If[Denominator[L[[i]]] == k, M = Append[M, L[[i]]]]]; Return[M]]
Earliest[k_] :=
Module[{i}, For[i = 1, Length[Denom[WF[i], k]] == 0, i++]; Return[i]]
Latest[k_] :=
Module[{i}, For[i = 1, Length[Denom[WF[i], k]] < EulerPhi[k], i++];
  Return[i]]
Table[Earliest[2 n + 1], {n, 1, 100}]
(* James Propp *)