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A325596 a(n) = Sum_{d|n} mu(n/d) * (-1)^(d + 1) * d. 3

%I

%S 1,-3,2,-2,4,-6,6,-4,6,-12,10,-4,12,-18,8,-8,16,-18,18,-8,12,-30,22,

%T -8,20,-36,18,-12,28,-24,30,-16,20,-48,24,-12,36,-54,24,-16,40,-36,42,

%U -20,24,-66,46,-16,42,-60,32,-24,52,-54,40,-24,36,-84,58,-16,60,-90,36,-32,48

%N a(n) = Sum_{d|n} mu(n/d) * (-1)^(d + 1) * d.

%C Moebius transform of A181983.

%F G.f.: Sum_{k>=1} mu(k) * x^k / (1 + x^k)^2.

%F G.f. A(x) satisfies: A(x) = x / (1 + x)^2 - Sum_{k>=2} A(x^k).

%F a(n) = phi(n) if n odd, phi(n) - 4*phi(n/2) if n even, where phi = A000010.

%F a(n) = A319997(n) - A319998(n).

%t a[n_] := Sum[MoebiusMu[n/d] (-1)^(d + 1) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]

%t a[n_] := If[OddQ[n], EulerPhi[n], EulerPhi[n] - 4 EulerPhi[n/2]]; Table[a[n], {n, 1, 65}]

%t nmax = 65; CoefficientList[Series[Sum[MoebiusMu[k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(-1)^(d+1)*d); \\ _Michel Marcus_, Sep 07 2019

%o (MAGMA) [&+[MoebiusMu(Floor(n/d))*(-1)^(d+1)*d:d in Divisors(n)]:n in [1..70]]; // _Marius A. Burtea_, Sep 07 2019

%Y Cf. A000010, A002129, A008683, A037225, A181983, A319997, A319998.

%K sign,mult

%O 1,2

%A _Ilya Gutkovskiy_, Sep 07 2019

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Last modified July 30 04:51 EDT 2021. Contains 346348 sequences. (Running on oeis4.)