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A279783 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. 3
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, 11, 11, 9, 12, 11, 10, 11, 12, 11, 13, 13, 13, 10, 11, 13, 12, 13, 14, 11, 13, 14, 13, 11, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The parameter k is the square of the Euclidean distance of the corresponding points to the origin in the "denominator, numerator" representation space.

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, ..., .

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

LINKS

Table of n, a(n) for n=1..81.

EXAMPLE

The point (1,0) lies closer to the origin than any other point in the first quadrant. Therefore a(1) = 1;

The next point is (1,1) because it lies closer to the origin. Therefore a(2) = 1;

The point (2,1) lies the next closest to the origin. Therefore a(3) = 2;

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

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

etc.

MATHEMATICA

nmax = 22; (* Maximum explorative denominator *)

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

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

fracsorted = SortBy[fracs, {(Numerator@#)^2 + (Denominator@#)^2 &,

    Denominator@# &}];

nmaxlimit = Floor[nmax^2/4] (*Safe limit for a correctly sorted sequence*);

Take[Denominator@fracsorted, nmaxlimit]

CROSSREFS

Cf. A279782.

Sequence in context: A061091 A196241 A086592 * A132663 A023964 A000267

Adjacent sequences:  A279780 A279781 A279782 * A279784 A279785 A279786

KEYWORD

nonn,frac

AUTHOR

Robert G. Wilson v and Andres Cicuttin, Dec 19 2016

STATUS

approved

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Last modified December 15 22:53 EST 2018. Contains 318157 sequences. (Running on oeis4.)