A328639 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-2)).

1, -5, -10, 5, -26, 50, -50, -5, 10, 130, -122, -50, -170, 250, 260, 5, -290, -50, -362, -130, 500, 610, -530, 50, 26, 850, -10, -250, -842, -1300, -962, -5, 1220, 1450, 1300, 50, -1370, 1810, 1700, 130, -1682, -2500, -1850, -610, -260, 2650, -2210, -50, 50, -130

Offset: 1

Author

Ilya Gutkovskiy, Oct 23 2019

Keywords

• sign
• mult

Comments

Dirichlet inverse of A065958.

Links

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

Crossrefs

Cf. A008683, A008836, A026424 (positions of negative terms), A046970, A065958, A323363, A328640.

Programs

Mathematica

a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^2, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 50}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^2 &], {n, 1, 50}]
f[p_, e_] := (-1)^e*(p^2 + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

PARI

a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA065958(n/d) * a(d).

a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^2, where lambda = A008836 and mu = A008683.

Multiplicative with a(p^e) = (-1)^e*(p^2 + 1). - Amiram Eldar, Nov 30 2020