A244396 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).

2, 5, 12, 21, 39, 54, 87, 117, 162, 204, 279, 333, 435, 516, 624, 732, 900, 1017, 1224, 1380, 1590, 1785, 2082, 2286, 2616, 2886, 3237, 3543, 4005, 4305, 4830, 5238, 5748, 6204, 6816, 7266, 8004, 8571, 9279, 9879, 10779, 11373, 12360, 13110, 14010, 14835

Offset

1,1

Links

Table of n, a(n) for n=1..46.

R. Tomás, From Farey sequences to resonance diagrams, Phys. Rev. ST Accel. Beams 17, 014001 - Published 29 January 2014.

R. Tomás, Asymptotic behavior of a series of Euler's Totient function times the cardinality of truncated Farey sequences, arXiv:1406.6991 [math.NT], 2014 (see Chapter 5, Evaluating ...).

Formula

a(n) = Sum_{k=1, n} A000010(k)*A049805(k, n).

a(n) = n^3/(6*zeta(3)) + O(n^2). (see (22) in Tomás link).

Mathematica

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 *)

Prog

(PARI) farey(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k); ); ); vecsort(Set(vf)); }
a(n) = my(row = farey(n)); sum(k=1, n, eulerphi(k)*vecsearch(row, 1/k));

Crossrefs

Cf. A000010, A049805, A006842, A006843.

Sequence in context: A307605 A079648 A080838 * A182993 A238741 A218904

Adjacent sequences: A244393 A244394 A244395 * A244397 A244398 A244399

Keyword

nonn

Author

Michel Marcus, Jun 27 2014

Status

approved