A279782 - OEIS

%I #23 Jul 20 2017 08:32:47

%S 0,1,1,1,2,1,3,1,2,3,1,4,1,2,3,5,4,1,3,5,1,6,2,5,4,1,5,3,7,1,2,7,3,4,

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

%U 5,8,11,1,2,3,7,4,12,5,11,6,1,9,7,11,5,8

%N Numerator 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) = 0;

%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) = 1;

%e The point (3,1) is the next closest to the origin. Therefore a(4)= 1;

%e The point (3,2) is the next closest to the origin. Therefore a(5)= 2;

%e etc.

%e The fractions 4/7 and 1/8 are respectively represented by the points (7,4) and (8,1), both points are at the same distance sqrt(65) from the origin, but 4/7 has smaller denominator than 1/8 hence a(17)=4 and a(18)=1.

%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[Numerator@fracsorted, nmaxlimit]

%Y Cf. A279783.

%K nonn,frac

%O 1,5

%A _Robert G. Wilson v_ and _Andres Cicuttin_, Dec 18 2016