A363551 - OEIS

%I #13 Jun 11 2023 02:57:25

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

%N Möbius function of rank 3: a(n) = lambda(n) = A008836(n) if n is cubefree and 0 otherwise.

%C First differs from A307420 at n = 63.

%C The Möbius function of rank 2 is the Möbius function (A008683).

%C Dirichlet inverse of A299406.

%H Amiram Eldar, <a href="/A363551/b363551.txt">Table of n, a(n) for n = 1..10000</a>

%H Masato Kobayashi, <a href="https://arxiv.org/abs/2108.01822">Möbius functions of higher rank and Dirichlet series</a>, arXiv:2108.01822 [math.NT], 2021.

%F a(n) = A008836(n) * A212793(n).

%F abs(a(n)) = A212793(n).

%F Multiplicative with a(p^e) = (-1)^e if e <= 2, and 0 otherwise.

%F Dirichlet g.f.: Product_{p prime} (1 - 1/p^s + 1/p^(2*s)) = zeta(2*s)*zeta(3*s)/(zeta(s)*zeta(6*s)).

%t f[p_, e_] := If[e < 3, (-1)^e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) f(e) = if(e < 3, (-1)^e, 0);

%o a(n) = vecprod(apply(f, factor(n)[,2]));

%Y Cf. A008683, A008836, A046099, A004709, A212793, A299406, A307420, A363552, A363553.

%Y Other generalizations of the Möbius function: A053864, A053865, A053981, A189021, A189022, A189023.

%K sign,mult,easy

%O 1

%A _Amiram Eldar_, Jun 10 2023