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A279066 Least prime q such that (q-p)/(r-q), where p are three consecutive primes, produces a new ratio <= 1, arranged by Farey fractions A038566/A038567.

%I A279066 #36 Apr 18 2023 09:44:24
%S A279066 5,3,31,23,8123,89,139,7963,337,409,199,797,45439,113,953,88547,293,
%T A279066 2633,1933,3643,137029,13381,523,2861,1381,1259,7621,7433,156157,3089,
%U A279066 546781,30911,1951,294563,1129,3229,285871,10369,14221,3651341,25819,3967,1669,6173,23473,51383
%N A279066 Least prime q such that (q-p)/(r-q), where p<q<r are three consecutive primes, produces a new ratio <= 1, arranged by Farey fractions A038566/A038567.
%C A279066 Almost a bisection of A275785 with only the term 5 being in both A279066 & A279067.
%C A279066 The union of A279066 & A279067 is A275785 with only 5 as a common term.
%C A279066 1/n = A179210(n).
%C A279066 Records: 5, 31, 8123, 45439, 88547, 137029, 156157, 546781, 3651341, 11931613, 16613347, 54636251, 72510257, 102626747, 148379059, 290018137, 847428851, 1165527283, 8232085373, 32592174133, 40113962921, ..., .
%H A279066 Andres Cicuttin and Robert G. Wilson v, <a href="/A279066/b279066.txt">Table of n, a(n) for n = 1..322</a>
%H A279066 Andres Cicuttin and Robert G. Wilson v, <a href="/A279066/a279066.txt">Table of n, Farey fraction = A038566/A038567 and a(n) for n=1..1000, or 0 if no such value is known</a>.
%e A279066 Row 1:        1/1                                       5
%e A279066 Row 2:        1/2                                       3
%e A279066 Row 3:     1/3  2/3                                 31      23
%e A279066 Row 4:     1/4  3/4                               8123      89
%e A279066 Row 5: 1/5 2/5  3/5 4/5                      139  7963     337    409
%e A279066 Row 6:     1/6  5/6                                199     797
%e A279066 Row 7:    1/7 .. 6/7                   45439 113   953   88547    293   2633
%e A279066 Row 8: 1/8 3/8  5/8 7/8                     1933  3643  137029  13381
%e A279066 etc.
%t A279066 f[n_] := Block[{p = 2, q = 3, r = 5}, While[q != n(r - q) + p, p = q; q = r; r = NextPrime@ r]; q]; Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}]; ff = Rest@ Reverse@ Sort[ Farey[25], Denominator[#2] < Denominator[#1] &]; f@# & /@ ff
%Y A279066 Cf. A275785, A279067.
%K A279066 nonn,easy
%O A279066 1,1
%A A279066 _Andres Cicuttin_ and _Robert G. Wilson v_, Dec 05 2016