A307420 - OEIS

%I #19 Dec 25 2022 02:11:59

%S 1,-1,-1,1,-1,1,-1,0,1,1,-1,-1,-1,1,1,0,-1,-1,-1,-1,1,1,-1,0,1,1,0,-1,

%T -1,-1,-1,0,1,1,1,1,-1,1,1,0,-1,-1,-1,-1,-1,1,-1,0,1,-1,1,-1,-1,0,1,0,

%U 1,1,-1,1,-1,1,-1,1,1,-1,-1,-1,1,-1,-1,0,-1,1,-1

%N Dirichlet g.f.: zeta(2*s) * zeta(3*s) / zeta(s).

%C Dirichlet convolution of A008836 and A010057.

%C Dirichlet convolution of A210826 and A010052.

%H Vaclav Kotesovec, <a href="/A307420/b307420.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletGeneratingFunction.html">Dirichlet Generating Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_series">Dirichlet series</a>.

%F Multiplicative with a(p^e) = 1 if e == 0 or 2 (mod 6), -1 if e == 1 (mod 6), and 0 otherwise. - _Amiram Eldar_, Dec 25 2022

%t Table[DivisorSum[n, Boole[IntegerQ[#^(1/3)]] * LiouvilleLambda[n/#]&], {n, 1, 100}]

%t Table[DivisorSum[n, Mod[DivisorSigma[0, #], 3, -1] * Boole[IntegerQ[Sqrt[n/#]]] &], {n, 1, 100}]

%t f[p_, e_] := Switch[Mod[e, 6], 0, 1, 1, -1, 2, 1, 3, 0, 4, 0, 5, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 25 2022 *)

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/((1+X)*(1-X^3)))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020

%Y Cf. A010052, A010057, A008836, A210826, A307425 (Dirichlet inverse).

%K sign,mult

%O 1

%A _Vaclav Kotesovec_, Apr 08 2019