A338547 a(n) = n^2 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^2.

1, -5, 8, -12, 24, -40, 48, -48, 72, -120, 120, -96, 168, -240, 192, -192, 288, -360, 360, -288, 384, -600, 528, -384, 600, -840, 648, -576, 840, -960, 960, -768, 960, -1440, 1152, -864, 1368, -1800, 1344, -1152, 1680, -1920, 1848, -1440, 1728, -2640, 2208, -1536, 2352, -3000

Offset

1,2

Comments

Moebius transform of A162395.

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) / (1 + x^k)^3.

G.f. A(x) satisfies: A(x) = x * (1 - x) / (1 + x)^3 - Sum_{k>=2} A(x^k).

Dirichlet g.f.: (1 - 2^(3 - s)) * zeta(s - 2) / zeta(s).

a(n) = J_2(n) if n odd, J_2(n) - 8 * J_2(n/2) if n even, where J_2 = A007434 (Jordan function J_2).

Multiplicative with a(2) = -5, a(2^e) = -3*2^(2*(e-1)) for e > 1, and a(p^e) = (p^2-1)*p^(2*(e-1)) for p > 2. - Amiram Eldar, Nov 15 2022

Mathematica

Table[n^2 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^2, {d, Divisors[n]}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - x^k)/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^2 - 1)*p^(2*(e - 1)); f[2, 1] = -5; f[2, e_] := -3*2^(2*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

Prog

(PARI) a(n) = n^2 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^2); \\ Michel Marcus, Nov 02 2020

Crossrefs

Cf. A007434, A008683, A162395, A321543, A325596, A338548, A338549.

Sequence in context: A138051 A026279 A260966 * A124434 A180930 A185729

Adjacent sequences: A338544 A338545 A338546 * A338548 A338549 A338550

Keyword

sign,mult

Author

Ilya Gutkovskiy, Nov 02 2020

Status

approved