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

Author

Anunay Kulshrestha, Jun 14 2012

Keywords

• nonn
• easy

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Wikipedia, Farey Sequence

Crossrefs

Similar to A133404 and A191607.

Partial sums of A023896.

Cf. A006842, A006843, A011755, A240877.

Programs

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 *)
Table[Total[Numerator[FareySequence[n]]],{n,50}] (* Harvey P. Dale, Apr 21 2025 *)

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