This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A078439 #26 Apr 30 2023 02:06:06
%S A078439 1,3,5,4,9,15,13,8,12,27,21,20,25,39,45,16,33,36,37,36,65,63,45,40,40,
%T A078439 75,36,52,57,135,61,32,105,99,117,48,73,111,125,72,81,195,85,84,108,
%U A078439 135,93,80,84,120,165,100,105,108,189,104,185,171,117,180,121,183,156,64
%N A078439 a(n) = Sum_{k=1..n} gcd(k,n)*mu(gcd(k,n))^2.
%H A078439 Amiram Eldar, <a href="/A078439/b078439.txt">Table of n, a(n) for n = 1..10000</a>
%F A078439 a(n) = Sum_{d|n} d*mu(d)^2*phi(n/d).
%F A078439 Multiplicative with a(p) = 2*p-1 and a(p^e) = 2*(p-1)*p^(e-1), e>1.
%F A078439 Dirichlet g.f.: zeta(s-1)^2 / (zeta(s) * zeta(2s-2)). - _Álvar Ibeas_, Mar 20 2015
%F A078439 Sum_{k=1..n} a(k) ~ 9 * n^2 * (2*log(n) + 4*gamma - 1 - 36*Zeta'(2)/Pi^2) / Pi^4, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Feb 01 2019
%t A078439 Table[Sum[d*MoebiusMu[d]^2*EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 100}] (* _Vaclav Kotesovec_, Feb 01 2019 *)
%t A078439 f[p_, e_] := If[e==1, 2*p-1, 2*(p-1)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 30 2023 *)
%o A078439 (PARI) vector(80, n, sumdiv(n, d, d*moebius(d)^2*eulerphi(n/d))) \\ _Michel Marcus_, Mar 20 2015
%o A078439 (Magma) [&+[Gcd(k,n)*MoebiusMu(Gcd(n,k))^2:k in [1..n]]:n in [1..70]]; // _Marius A. Burtea_, Sep 15 2019
%Y A078439 Cf. A001620, A000010, A008683, A063659.
%K A078439 mult,nonn
%O A078439 1,2
%A A078439 _Vladeta Jovovic_, Dec 31 2002