A338548 a(n) = n^3 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^3.

1, -9, 26, -56, 124, -234, 342, -448, 702, -1116, 1330, -1456, 2196, -3078, 3224, -3584, 4912, -6318, 6858, -6944, 8892, -11970, 12166, -11648, 15500, -19764, 18954, -19152, 24388, -29016, 29790, -28672, 34580, -44208, 42408, -39312, 50652, -61722, 57096, -55552, 68920, -80028

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - 4*x^k + x^(2*k)) / (1 + x^k)^4.

G.f. A(x) satisfies: A(x) = x * (1 - 4*x + x^2) / (1 + x)^4 - Sum_{k>=2} A(x^k).

Dirichlet g.f.: (1 - 2^(4 - s)) * zeta(s - 3) / zeta(s).

a(n) = J_3(n) if n odd, J_3(n) - 16 * J_3(n/2) if n even, where J_3 = A059376 (Jordan function J_3).

Multiplicative with a(2) = -9, a(2^e) = -7*2^(3*(e-1)) for e > 1, and a(p^e) = (p^3-1)*p^(3*(e-1)) for p > 2. - Amiram Eldar, Nov 15 2022

Mathematica

Table[n^3 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^3, {d, Divisors[n]}], {n, 1, 42}]
nmax = 42; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - 4 x^k + x^(2 k))/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^3 - 1)*p^(3*(e - 1)); f[2, 1] = -9; f[2, e_] := -7*2^(3*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

Prog

(PARI) a(n) = n^3 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^3); \\ Michel Marcus, Nov 02 2020

Crossrefs

Cf. A008683, A059376, A138503, A325596, A338547, A338549.

Sequence in context: A081267 A052153 A154560 * A249275 A270695 A048468

Adjacent sequences: A338545 A338546 A338547 * A338549 A338550 A338551

Keyword

sign,mult

Author

Ilya Gutkovskiy, Nov 02 2020

Status

approved