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A078439 a(n) = Sum_{k=1..n} gcd(k,n)*mu(gcd(k,n))^2. 2

%I

%S 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 75,36,52,57,135,61,32,105,99,117,48,73,111,125,72,81,195,85,84,108,

%U 135,93,80,84,120,165,100,105,108,189,104,185,171,117,180,121,183,156,64

%N a(n) = Sum_{k=1..n} gcd(k,n)*mu(gcd(k,n))^2.

%H Amiram Eldar, <a href="/A078439/b078439.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} d*mu(d)^2*phi(n/d).

%F Multiplicative with a(p) = 2*p-1 and a(p^e) = 2*(p-1)*p^(e-1), e>1.

%F Dirichlet g.f.: zeta(s-1)^2 / (zeta(s) * zeta(2s-2)). - _Álvar Ibeas_, Mar 20 2015

%F 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 Table[Sum[d*MoebiusMu[d]^2*EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 100}] (* _Vaclav Kotesovec_, Feb 01 2019 *)

%o (PARI) vector(80, n, sumdiv(n, d, d*moebius(d)^2*eulerphi(n/d))) \\ _Michel Marcus_, Mar 20 2015

%o (MAGMA) [&+[Gcd(k,n)*MoebiusMu(Gcd(n,k))^2:k in [1..n]]:n in [1..70]]; // _Marius A. Burtea_, Sep 15 2019

%Y Cf. A008683, A000010, A063659.

%K mult,nonn

%O 1,2

%A _Vladeta Jovovic_, Dec 31 2002

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Last modified November 26 07:49 EST 2020. Contains 338632 sequences. (Running on oeis4.)