A178031 -id:A178031 - cp's OEIS frontend

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

N. J. A. Sloane, Dec 15 2010

nonn

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

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

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

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 *)

A178047 Consider the Farey tree A049455/A049456; a(n) = row at which the denominator n first appears (assumes first row is labeled row 0).

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 8, 8, 7, 8, 8, 8, 9, 8, 8, 8, 9, 8, 8, 8, 8, 8, 9, 8, 8, 9, 9, 9, 10, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 9, 10, 10, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 9, 11, 10, 10, 10, 10, 10, 11

Offset: 1

N. J. A. Sloane, Dec 16 2010

nonn

Equals A178031 - 1. See that entry for further information.

Richard J. Mathar, The Kepler binary tree of reduced fractions, 2017.

Cf. A295783 (frequencies of this).

A378568 Lowest weight of rational fraction with denominator n.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 3, 2, 6, 2, 6, 2, 3, 2, 7, 2, 7, 2, 3, 2, 8, 2, 4, 2, 3, 2, 8, 2, 8, 2, 3, 2, 4, 2, 9, 2, 3, 2, 9, 2, 9, 2, 3, 2, 9, 2, 5, 2, 3, 2, 10, 2, 4, 2, 3, 2, 10, 2, 10, 2, 3, 2, 4, 2, 10, 2, 3, 2, 10, 2, 10, 2, 3, 2, 5, 2, 10, 2, 3, 2, 11, 2

Offset: 1

Jeffrey Shallit, Dec 01 2024

nonn

The weight wt(x) of a rational number x is defined to be the sum of the partial quotients in the continued fraction expansion of x. For example, 5/14 = [0,2,1,4], so wt(5/14) = 7. Here a(n) is the minimum, over all m, 1<=m

It is conjectured by Kravitz and Sah that a(n) = O(log n).

			For n = 23, we have a(23) = 8 because 5/23 = [0,4,1,1,2] with weight 8, and this is the smallest over all fractions m/23 with 1<=m<23.

N. Kravitz and A. Sah, Linear extension numbers of n-element posets, Order 38 (2021), 49-66.
M. Shrader-Frechette, Modified Farey sequences and continued fractions, Math. Magazine 54 (1991), 60-63.

Cf. A178031 (same but for irreducible fractions only).

a(n)=if(n==1, return(1)); my(r=oo,t); for(m=1,n-1, t=vecsum(contfrac(m/n)); if(tCharles R Greathouse IV, Dec 01 2024

a(n) = min_{d|n, d>1} A178031(d) for n>1. - Andrey Zabolotskiy, Dec 01 2024

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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).

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A178047 Consider the Farey tree A049455/A049456; a(n) = row at which the denominator n first appears (assumes first row is labeled row 0).

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A378568 Lowest weight of rational fraction with denominator n.

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