A328729 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(2*s)).

1, -1, -2, -2, -4, 2, -6, 0, -3, 4, -10, 4, -12, 6, 8, 0, -16, 3, -18, 8, 12, 10, -22, 0, -5, 12, 0, 12, -28, -8, -30, 0, 20, 16, 24, 6, -36, 18, 24, 0, -40, -12, -42, 20, 12, 22, -46, 0, -7, 5, 32, 24, -52, 0, 40, 0, 36, 28, -58, -16, -60, 30, 18, 0, 48, -20, -66, 32, 44, -24

Offset

1,3

Comments

Dirichlet inverse of A206369.

Links

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

Formula

a(1) = 1; a(n) = -Sum_{d|n, d<n} A206369(n/d) * a(d).

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

a(n) = Sum_{d|n} A008966(n/d) * A055615(d).

a(n) = Sum_{d|n} A271102(n/d) * A023900(d).

Multiplicative with a(p^e) = 2 - p - e if e < 3, and 0 otherwise. - Amiram Eldar, Dec 02 2020

Mathematica

a[1] = 1; a[n_] := -Sum[(n/d) DivisorSum[n/d, LiouvilleLambda[#]/# &] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 70}]
Table[DivisorSum[n, MoebiusMu[n/#]^2 MoebiusMu[#] # &], {n, 1, 70}]
f[p_, e_] := If[e < 3, -p - e + 2, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Prog

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

Crossrefs

Cf. A008683, A008966, A023900, A046099 (positions of 0's), A046692, A055615, A206369, A271102.

Sequence in context: A179013 A090397 A339176 * A349441 A329733 A289624

Adjacent sequences: A328726 A328727 A328728 * A328730 A328731 A328732

Keyword

sign,mult

Author

Ilya Gutkovskiy, Oct 26 2019

Status

approved