A307421 - OEIS

%I #22 Dec 25 2022 02:10:10

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

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

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

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

%C Dirichlet convolution of A008966 and A010057.

%H Vaclav Kotesovec, <a href="/A307421/b307421.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A307421/a307421.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 a(n) = abs(A210826(n)).

%F Sum_{k=1..n} a(k) ~ 6*zeta(3)*n/Pi^2 + zeta(1/3)*n^(1/3)/zeta(2/3).

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

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

%t f[p_, e_] := If[Mod[e, 3] == 2, 0, 1]; 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. A010057, A008966, A210826, A299406.

%K nonn,mult

%O 1

%A _Vaclav Kotesovec_, Apr 08 2019