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

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

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