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%I #18 May 29 2020 07:49:28
%S 2,5,12,21,39,54,87,117,162,204,279,333,435,516,624,732,900,1017,1224,
%T 1380,1590,1785,2082,2286,2616,2886,3237,3543,4005,4305,4830,5238,
%U 5748,6204,6816,7266,8004,8571,9279,9879,10779,11373,12360,13110,14010,14835
%N a(n) = Sum_{k=1, n} phi(k)*index(k, n), with phi(k) the Euler totient A000010(k) and index(k,n) the position of 1/k in the n-th row of the Farey sequence of order k, A049805(n,k).
%H R. Tomás, <a href="http://dx.doi.org/10.1103/PhysRevSTAB.17.014001">From Farey sequences to resonance diagrams</a>, Phys. Rev. ST Accel. Beams 17, 014001 - Published 29 January 2014.
%H R. Tomás, <a href="http://arxiv.org/abs/1406.6991">Asymptotic behavior of a series of Euler's Totient function times the cardinality of truncated Farey sequences</a>, arXiv:1406.6991 [math.NT], 2014 (see Chapter 5, Evaluating ...).
%F a(n) = Sum_{k=1, n} A000010(k)*A049805(k, n).
%F a(n) = n^3/(6*zeta(3)) + O(n^2). (see (22) in Tomás link).
%t a[n_] := With[{f = FareySequence[n]}, Sum[EulerPhi[k] FirstPosition[f, 1/k ][[1]], {k, 1, n}]]; Array[a, 50] (* _Jean-François Alcover_, Sep 26 2018 *)
%o (PARI) farey(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k););); vecsort(Set(vf));}
%o a(n) = my(row = farey(n)); sum(k=1, n, eulerphi(k)*vecsearch(row, 1/k));
%Y Cf. A000010, A049805, A006842, A006843.
%K nonn
%O 1,1
%A _Michel Marcus_, Jun 27 2014
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