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

0,3

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.

Links

Table of n, a(n) for n=0..100.

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 *)

Crossrefs

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

Sequence in context: A038809 A337496 A078342 * A107325 A003050 A070868

Adjacent sequences: A177900 A177901 A177902 * A177904 A177905 A177906

Keyword

nonn

Author

N. J. A. Sloane, Dec 15 2010

Status

approved