A326415 Dirichlet g.f.: zeta(2*s) / zeta(s)^3.

1, -3, -3, 4, -3, 9, -3, -4, 4, 9, -3, -12, -3, 9, 9, 4, -3, -12, -3, -12, 9, 9, -3, 12, 4, 9, -4, -12, -3, -27, -3, -4, 9, 9, 9, 16, -3, 9, 9, 12, -3, -27, -3, -12, -12, 9, -3, -12, 4, -12, 9, -12, -3, 12, 9, 12, 9, 9, -3, 36, -3, 9, -12, 4, 9, -27, -3, -12, 9, -27, -3, -16, -3, 9, -12

Offset

1,2

Comments

Moebius transform applied twice to A008836.

Dirichlet inverse of A048691.

Links

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

Ilya Gutkovskiy, Scatter plot of partial sums of A326415 up to n=10000.

Formula

a(n) = Sum_{d|n} mu(n/d) * (-1)^bigomega(d) * 2^omega(d), where mu = A008683, bigomega = A001222 and omega = A001221.

a(1) = 1; a(n) = -Sum_{d|n, d<n} tau((n/d)^2) * a(d), where tau = A000005.

a(n) = Sum_{d|n} A008836(n/d) * A007427(d).

a(n) = Sum_{d|n} A010052(n/d) * A007428(d).

Multiplicative with a(p^e) = -3 if e = 1, and 4*(-1)^e otherwise. - Amiram Eldar, Oct 26 2020

b(n) = abs( a(n) ) is multiplicative with b(p) = 3 and b(p^e) = 4 for e > 1 and prime p. Its Dirichlet g.f. is: zeta(s)^3 / zeta(2*s)^2. - Werner Schulte, Jan 18 2023

Mathematica

Table[Sum[MoebiusMu[n/d] (-1)^PrimeOmega[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSigma[0, (n/d)^2] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 75}]
f[p_, e_] := If[e == 1, -3, (-1)^e*4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

Crossrefs

Cf. A000005, A001221, A001222, A007427, A007428, A008836, A010052, A034444, A046951, A048691, A158522.

Sequence in context: A267048 A350502 A326401 * A343443 A351348 A246011

Adjacent sequences: A326412 A326413 A326414 * A326416 A326417 A326418

Keyword

sign,mult,easy

Author

Ilya Gutkovskiy, Oct 18 2019

Status

approved