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%I #23 Jul 20 2017 08:32:55
%S 1,1,2,3,3,4,4,5,5,5,6,5,7,7,7,6,7,8,8,7,9,7,9,8,9,10,9,10,8,11,11,9,
%T 11,11,9,12,11,10,11,12,11,13,13,13,10,11,13,12,13,14,11,13,14,13,11,
%U 14,15,15,13,15,13,16,12,16,13,15,14,16,15,13,17,17,17,16,17,13,17,14,17,18,16
%N Denominator of Farey fractions sorted according to increasing k, with k = numerator^2 + denominator^2. Fractions with same k are sorted in order of increasing denominator.
%C The parameter k is the square of the Euclidean distance of the corresponding points to the origin in the "denominator, numerator" representation space.
%C The fractions in order begin: 0/1, 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 1/6, 4/5, 1/7, 2/7, 3/7, 5/6, ..., .
%C Note that the fraction 2/4 is not in the above since it can be reduced to 1/2.
%e The point (1,0) lies closer to the origin than any other point in the first quadrant. Therefore a(1) = 1;
%e The next point is (1,1) because it lies closer to the origin. Therefore a(2) = 1;
%e The point (2,1) lies the next closest to the origin. Therefore a(3) = 2;
%e The point (3,1) is the next closest to the origin. Therefore a(4)= 3;
%e The point (3,2) is the next closest to the origin. Therefore a(5)= 3;
%e etc.
%t nmax = 22; (* Maximum explorative denominator *)
%t fracs = Sort@Union@Flatten@Table[a/b, {b, nmax}, {a, 0, b}];
%t (* Sorting generated fractions according to increasing Euclidean distance first, and then by increasing denominator *)
%t fracsorted = SortBy[fracs, {(Numerator@#)^2 + (Denominator@#)^2 &,
%t Denominator@# &}];
%t nmaxlimit = Floor[nmax^2/4] (*Safe limit for a correctly sorted sequence*);
%t Take[Denominator@fracsorted, nmaxlimit]
%Y Cf. A279782.
%K nonn,frac
%O 1,3
%A _Robert G. Wilson v_ and _Andres Cicuttin_, Dec 19 2016
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