A328729 - OEIS

%I #10 Dec 02 2020 03:16:27

%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,

%T 12,0,12,-28,-8,-30,0,20,16,24,6,-36,18,24,0,-40,-12,-42,20,12,22,-46,

%U 0,-7,5,32,24,-52,0,40,0,36,28,-58,-16,-60,30,18,0,48,-20,-66,32,44,-24

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

%C Dirichlet inverse of A206369.

%H Amiram Eldar, <a href="/A328729/b328729.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

%F Multiplicative with a(p^e) = 2 - p - e if e < 3, and 0 otherwise. - _Amiram Eldar_, Dec 02 2020

%t a[1] = 1; a[n_] := -Sum[(n/d) DivisorSum[n/d, LiouvilleLambda[#]/# &] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 70}]

%t Table[DivisorSum[n, MoebiusMu[n/#]^2 MoebiusMu[#] # &], {n, 1, 70}]

%t 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 *)

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)^2*moebius(d)*d); \\ _Michel Marcus_, Dec 02 2020

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

%K sign,mult

%O 1,3

%A _Ilya Gutkovskiy_, Oct 26 2019