Re: [math-fun] q. re Farey-ish fractions with weighted mediants
I haven't looked at "Farey-depth". Maybe someone on this list could work out the initial terms of the two sequences in 2.?
Earliest occurrences of odd denominators 3,5,7,...,201: 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 Latest occurrences of odd denominators 3,5,7,...,29: 1,3,3,4,5,6,7,8,9,10,11,12,13,14,15 (The glitch in the second term of the second of these sequences 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.) Here's the Mathematica code I used: 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}] Jim
I don't see where the restriction to odd denominators comes from. --Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp [jpropp@cs.uml.edu] Sent: Tuesday, December 14, 2010 10:15 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] q. re Farey-ish fractions with weighted mediants
I haven't looked at "Farey-depth". Maybe someone on this list could work out the initial terms of the two sequences in 2.?
Earliest occurrences of odd denominators 3,5,7,...,201: 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 Latest occurrences of odd denominators 3,5,7,...,29: 1,3,3,4,5,6,7,8,9,10,11,12,13,14,15 (The glitch in the second term of the second of these sequences 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.) Here's the Mathematica code I used: 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}] Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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James Propp -
Schroeppel, Richard