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
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
Programs
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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
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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