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 A325596 a(n) = Sum_{d|n} mu(n/d) * (-1)^(d + 1) * d. 4
 1, -3, 2, -2, 4, -6, 6, -4, 6, -12, 10, -4, 12, -18, 8, -8, 16, -18, 18, -8, 12, -30, 22, -8, 20, -36, 18, -12, 28, -24, 30, -16, 20, -48, 24, -12, 36, -54, 24, -16, 40, -36, 42, -20, 24, -66, 46, -16, 42, -60, 32, -24, 52, -54, 40, -24, 36, -84, 58, -16, 60, -90, 36, -32, 48 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Moebius transform of A181983. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 FORMULA G.f.: Sum_{k>=1} mu(k) * x^k / (1 + x^k)^2. G.f. A(x) satisfies: A(x) = x / (1 + x)^2 - Sum_{k>=2} A(x^k). a(n) = phi(n) if n odd, phi(n) - 4*phi(n/2) if n even, where phi = A000010. a(n) = A319997(n) - A319998(n). Multiplicative with a(2) = -3, a(2^e) = -2^(e-1) for e > 1, and a(p^e) = (p-1)*p^(e-1) for p > 2. - Amiram Eldar, Nov 15 2022 MATHEMATICA a[n_] := Sum[MoebiusMu[n/d] (-1)^(d + 1) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 65}] a[n_] := If[OddQ[n], EulerPhi[n], EulerPhi[n] - 4 EulerPhi[n/2]]; Table[a[n], {n, 1, 65}] nmax = 65; CoefficientList[Series[Sum[MoebiusMu[k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest f[p_, e_] := (p - 1)*p^(e - 1); f[2, 1] = -3; f[2, e_] := -2^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *) PROG (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(-1)^(d+1)*d); \\ Michel Marcus, Sep 07 2019 (Magma) [&+[MoebiusMu(Floor(n/d))*(-1)^(d+1)*d:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Sep 07 2019 CROSSREFS Cf. A000010, A002129, A008683, A037225, A181983, A319997, A319998. Sequence in context: A308194 A245572 A344985 * A254876 A249159 A230871 Adjacent sequences: A325593 A325594 A325595 * A325597 A325598 A325599 KEYWORD sign,mult AUTHOR Ilya Gutkovskiy, Sep 07 2019 STATUS approved

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Last modified January 30 05:43 EST 2023. Contains 359939 sequences. (Running on oeis4.)