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A071708 Numerator of Sum_{k=1..n} phi(k)/k.

%I A071708 #21 Sep 18 2022 02:01:59
%S A071708 1,3,13,8,52,19,163,361,1223,1307,16477,5749,83977,88267,280817,
%T A071708 147916,2754812,2839897,58552633,60492571,63263911,65468386,
%U A071708 1612469468,549883871,579629587,596790577,1864736021,1912541636,58587007624,59449633388,1939984033478
%N A071708 Numerator of Sum_{k=1..n} phi(k)/k.
%D A071708 József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section I.24, page 27.
%D A071708 Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, 1963.
%H A071708 G. C. Greubel, <a href="/A071708/b071708.txt">Table of n, a(n) for n = 1..1000</a>
%F A071708 Also numerator of Sum_{i=1..n} (mu(i)/i)*floor(n/i). - _Ridouane Oudra_, Nov 26 2019
%F A071708 a(n)/A072155(n) ~ (6/Pi^2) * n + O(log(n)^(2/3)*log(log(n))^(4/3)). - _Amiram Eldar_, Sep 18 2022
%e A071708 1, 3/2, 13/6, 8/3, 52/15, 19/5, 163/35, 361/70, 1223/210, ...
%p A071708 with(numtheory); seq(numer(add(phi(k)/k, k = 1..n)), n = 1..35); # _G. C. Greubel_, Aug 25 2019
%t A071708 Table[Sum[EulerPhi[k]/k, {k, n}], {n,35}]//Numerator (* _G. C. Greubel_, Aug 25 2019 *)
%o A071708 (PARI) a(n) = numerator(sum(k=1,n, eulerphi(k)/k));
%o A071708 vector(35, n, a(n)) \\ _G. C. Greubel_, Aug 25 2019
%o A071708 (Magma) [Numerator( &+[EulerPhi(k)/k: k in [1..n]] ): n in [1..35]]; // _G. C. Greubel_, Aug 25 2019
%o A071708 (Sage) [numerator( sum(euler_phi(k)/k for k in (1..n)) ) for n in (1..35)] # _G. C. Greubel_, Aug 25 2019
%o A071708 (GAP) List([1..35], n-> NumeratorRat( Sum([1..n], k-> Phi(k)/k) ) ); # _G. C. Greubel_, Aug 25 2019
%Y A071708 Cf. A072155 (denominators), A000010, A059956.
%K A071708 nonn,frac
%O A071708 1,2
%A A071708 _N. J. A. Sloane_, Jun 28 2002