A213544 Sum of numerators of Farey Sequence of order n.

1, 2, 5, 9, 19, 25, 46, 62, 89, 109, 164, 188, 266, 308, 368, 432, 568, 622, 793, 873, 999, 1109, 1362, 1458, 1708, 1864, 2107, 2275, 2681, 2801, 3266, 3522, 3852, 4124, 4544, 4760, 5426, 5768, 6236, 6556, 7376, 7628, 8531, 8971, 9511, 10017, 11098, 11482

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Wikipedia, Farey Sequence

Formula

a(n) = Sum_{k=1..n} A023896(k).

a(n) = A240877(n)/2. - Robert G. Wilson v, Apr 15 2014

a(n) ~ n^3/Pi^2 - Jean-François Alcover, Dec 29 2014

a(n) = (A011755(n)+1)/2. - Chai Wah Wu, Apr 04 2022

Example

For n = 3, the Farey Sequence is 0/1, 1/3, 1/2, 2/3, 1/1. Thus a(3) = 0 + 1 + 1 + 2 + 1 = 5.

Maple

with(numtheory):
b:= n-> `if`(n=1, 1, n*phi(n)/2):
a:= proc(n) option remember; b(n) +`if`(n>1, a(n-1), 0) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 14 2012

Mathematica

Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Table[ Total[ Numerator[ Farey[ n]]], {n, 0, 53}] (* Robert G. Wilson v, Apr 15 2014 *)
a[n_] := Sum[If[CoprimeQ[j, k], j, 0], {k, 1, n}, {j, 1, k}]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Dec 29 2014 *)

Crossrefs

Similar to A133404 and A191607.

Partial sums of A023896.

Cf. A006842, A006843, A011755, A240877.

Sequence in context: A152546 A286713 A342013 * A265482 A085410 A073118

Adjacent sequences: A213541 A213542 A213543 * A213545 A213546 A213547

Keyword

nonn,easy

Author

Anunay Kulshrestha, Jun 14 2012

Status

approved