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A213544 Sum of numerators of Farey Sequence of order n. 6

%I #27 Apr 04 2022 12:49:22

%S 1,2,5,9,19,25,46,62,89,109,164,188,266,308,368,432,568,622,793,873,

%T 999,1109,1362,1458,1708,1864,2107,2275,2681,2801,3266,3522,3852,4124,

%U 4544,4760,5426,5768,6236,6556,7376,7628,8531,8971,9511,10017,11098,11482

%N Sum of numerators of Farey Sequence of order n.

%H Alois P. Heinz, <a href="/A213544/b213544.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Farey_sequence">Farey Sequence</a>

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

%F a(n) = A240877(n)/2. - _Robert G. Wilson v_, Apr 15 2014

%F a(n) ~ n^3/Pi^2 - _Jean-François Alcover_, Dec 29 2014

%F a(n) = (A011755(n)+1)/2. - _Chai Wah Wu_, Apr 04 2022

%e 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.

%p with(numtheory):

%p b:= n-> `if`(n=1, 1, n*phi(n)/2):

%p a:= proc(n) option remember; b(n) +`if`(n>1, a(n-1), 0) end:

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Jun 14 2012

%t 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 *)

%t 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 *)

%Y Similar to A133404 and A191607.

%Y Partial sums of A023896.

%Y Cf. A006842, A006843, A011755, A240877.

%K nonn,easy

%O 1,2

%A _Anunay Kulshrestha_, Jun 14 2012

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)