A328640 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-3)).

1, -9, -28, 9, -126, 252, -344, -9, 28, 1134, -1332, -252, -2198, 3096, 3528, 9, -4914, -252, -6860, -1134, 9632, 11988, -12168, 252, 126, 19782, -28, -3096, -24390, -31752, -29792, -9, 37296, 44226, 43344, 252, -50654, 61740, 61544, 1134, -68922, -86688, -79508, -11988, -3528

Offset

1,2

Comments

Dirichlet inverse of A065959.

Links

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

Formula

a(1) = 1; a(n) = -Sum_{d|n, d<n} A065959(n/d) * a(d).

a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^3, where lambda = A008836 and mu = A008683.

Multiplicative with a(p^) = (-1)^e*(p^3+1). - Amiram Eldar, Dec 03 2022

Mathematica

a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^3, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 45}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^3 &], {n, 1, 45}]
f[p_, e_] := (-1)^e*(p^3+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

Prog

(PARI) a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^3)} \\ Andrew Howroyd, Oct 25 2019

Crossrefs

Cf. A008683, A008836, A026424 (positions of negative terms), A063453, A065959, A323363, A328639.

Sequence in context: A075539 A225300 A053825 * A351266 A369721 A369759

Adjacent sequences: A328637 A328638 A328639 * A328641 A328642 A328643

Keyword

sign,mult

Author

Ilya Gutkovskiy, Oct 23 2019

Status

approved