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A319998 a(n) = Sum_{d|n, d is even} mu(n/d)*d, where mu(n) is Moebius function A008683.

%I A319998 #18 Nov 12 2022 05:25:27
%S A319998 0,2,0,2,0,4,0,4,0,8,0,4,0,12,0,8,0,12,0,8,0,20,0,8,0,24,0,12,0,16,0,
%T A319998 16,0,32,0,12,0,36,0,16,0,24,0,20,0,44,0,16,0,40,0,24,0,36,0,24,0,56,
%U A319998 0,16,0,60,0,32,0,40,0,32,0,48,0,24,0,72,0,36,0,48,0,32,0,80,0,24,0,84,0,40,0,48,0,44,0,92,0,32,0,84,0,40,0,64,0,48,0
%N A319998 a(n) = Sum_{d|n, d is even} mu(n/d)*d, where mu(n) is Moebius function A008683.
%H A319998 Antti Karttunen, <a href="/A319998/b319998.txt">Table of n, a(n) for n = 1..20000</a>
%F A319998 a(n) = Sum_{d|n} A059841(d)*A008683(n/d)*d.
%F A319998 a(n) = A000010(n) - A319997(n).
%F A319998 a(2n) = 2*A000010(n), a(2n+1) = 0.
%F A319998 G.f.: Sum_{k>=1} 2*mu(k)*x^(2*k)/(1 - x^(2*k))^2. - _Ilya Gutkovskiy_, Nov 02 2018
%F A319998 Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/(2*Pi^2) = 0.151981... . - _Amiram Eldar_, Nov 12 2022
%t A319998 Rest[CoefficientList[Series[Sum[2*MoebiusMu[k]*x^(2*k)/(1 - x^(2*k))^2, {k, 1, 100}], {x, 0, 100}], x]] (* _Vaclav Kotesovec_, Nov 03 2018 *)
%o A319998 (PARI) A319998(n) = sumdiv(n,d,(!(d%2))*moebius(n/d)*d);
%o A319998 (PARI) A319998(n) = if(n%2, 0, 2*eulerphi(n/2));
%Y A319998 Cf. A000010, A008683, A059841, A140434, A146076, A319997.
%K A319998 nonn
%O A319998 1,2
%A A319998 _Antti Karttunen_, Oct 31 2018