A280073 - OEIS

%I #23 Jul 20 2017 08:33:01

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

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

%U 4,2,1,9,7,5,3,1,11,10,9,8,7,6,5,4,3,2,1,11,7,5,1,12,11,9,8,7,6,4,3,2,1,11,9,7

%N Numerator of Farey fractions sorted according to increasing k, with k = numerator + denominator. Fractions with same k are sorted in order of increasing denominator.

%C The parameter k is the Manhattan 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, 1/5, 3/4, 2/5, 1/6, 3/5, 1/7, 4/5, 2/7, 1/8, ..., .

%C Note that the fraction 2/4 is not in the above since it can be reduced to 1/2.

%t nmax = 25;

%t (* fracs are fractions represented in the triangle with vertices

%t (0,1),(1,nmax) and (nmax,nmax) *)

%t fracs = Sort@Union@Flatten@Table[a/b, {b, nmax}, {a, 0, b}];

%t (* Sorting generated fractions according to increasing Manhattan distance first, and then by increasing denominator *)

%t fracsorted =

%t   SortBy[fracs, {Numerator@# + Denominator@# &, Denominator@# &}];

%t nmaxlimit = Floor[(1/6)* nmax^2]; (* Safe limit for a correctly sorted sequence since asymptotically half of the generated fractions can be properly sorted according to Manhattan distance *)

%t Take[Numerator@fracsorted, nmaxlimit]

%Y Cf. A279782, A279783.

%K nonn,frac

%O 1,5

%A _Robert G. Wilson v_ and _Andres Cicuttin_, Dec 25 2016