A307425 - OEIS

%I #23 Feb 07 2023 08:22:57

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

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

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

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

%C Dirichlet convolution of A212793 and A271102.

%H Vaclav Kotesovec, <a href="/A307425/b307425.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A307425/a307425.jpg">Graph - the asymptotic ratio</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 Sum_{k=1..n} a(k) ~ 6*n / (Pi^2 * zeta(3)).

%F Multiplicative with a(p) = 1, a(p^e) = -1 if e = 3 or 4, and 0 if e = 2 or e >= 5. - _Amiram Eldar_, Dec 25 2022

%t nmax = 100; A271102 = Table[DivisorSum[n, Abs[MoebiusMu[#]]*MoebiusMu[n/#] &], {n, 1, nmax}]; Table[DivisorSum[n, Boole[Max[FactorInteger[#][[All, 2]]] < 3] * A271102[[n/#]] &], {n, 1, nmax}]

%t f[p_, e_] := Switch[e, 1, 1, 3, -1, 4, -1, _, 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+X)*(1-X^3))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020

%Y Cf. A056624, A210826, A212793, A271102, A299406, A307420 (Dirichlet inverse).

%K sign,mult

%O 1

%A _Vaclav Kotesovec_, Apr 08 2019