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A211177 Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

%I A211177 #32 Oct 14 2022 09:26:33
%S A211177 -1,0,-1,0,-1,1,1,1,1,5,19,17,29,13,21,13,47,181,503,593,533,121,1259,
%T A211177 1457,6889,7549,7109,7769,52403,59333,11497,6095,29089,61643,59333,
%U A211177 63953,62413,7277,21061,2777,10877,11647,3809,3963,1438,271,3064,51439,7217,7493
%N A211177 Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.
%H A211177 Amiram Eldar, <a href="/A211177/b211177.txt">Table of n, a(n) for n = 1..10000</a>
%H A211177 Olivier Bordellès and Benoit Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Bordelles/bord14.html">An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions</a>, J. Int. Seq., Vol. 16 (2013) Article 13.6.3.
%H A211177 László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
%F A211177 a(n)/A211178(n) = c*log(n) + O(1) with a suitable constant c (see ref).
%F A211177 The constant above is c = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - _Amiram Eldar_, Nov 20 2020
%F A211177 More accurately, a(n)/A211178(n) ~ (A/3) * (log(n) + gamma - B - 8*log(2)/3) + O(log(n)^(5/3)/n), where A = zeta(2)*zeta(3)/zeta(6) (A082695), gamma is Euler's constant (A001620), and B = Sum_{p prime} log(p)/(p^2-p+1) (A085609) (Bordellès and Cloitre, 2013; Tóth, 2017). - _Amiram Eldar_, Oct 14 2022
%e A211177 Fractions begin with -1, 0, -1/2, 0, -1/4, 1/4, 1/12, 1/3, 1/6, 5/12, 19/60, 17/30, ...
%t A211177 Numerator @ Accumulate[Table[(-1)^k/EulerPhi[k], {k, 1, 50}]] (* _Amiram Eldar_, Nov 20 2020 *)
%o A211177 (PARI) a(n)=numerator(sum(k=1,n,(-1)^k/eulerphi(k)))
%Y A211177 Cf. A000010, A028415, A211178 (denominators).
%Y A211177 Cf. A001620, A082695, A085609.
%K A211177 sign,frac
%O A211177 1,10
%A A211177 _Benoit Cloitre_, Feb 01 2013
%E A211177 More terms from _Amiram Eldar_, Nov 20 2020