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Dirichlet inverse of A354354, characteristic function of numbers that are neither multiples of 2 nor 3.
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%I #19 Dec 27 2022 02:30:08

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

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

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

%N Dirichlet inverse of A354354, characteristic function of numbers that are neither multiples of 2 nor 3.

%C Multiplicative because A354354 is.

%H Antti Karttunen, <a href="/A355688/b355688.txt">Table of n, a(n) for n = 1..100000</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A354354(n/d) * a(d).

%F a(n) = A008683(n) * A354354(n). [As the latter is fully multiplicative] - _Antti Karttunen_, Dec 20 2022

%F From _Amiram Eldar_, Dec 27 2022: (Start)

%F Multiplicative with a(p^e) = 0 if p <= 3 or e >= 2, and -1 otherwise.

%F Dirichlet g.f.: (2^s)*(3^s)/(zeta(s)*(2^s-1)*(3^s-1)). (End)

%t a[n_] := If[GCD[n, 6] == 1, MoebiusMu[n], 0]; Array[a, 100] (* _Amiram Eldar_, Dec 27 2022 *)

%o (PARI)

%o A354354(n) = ((n%2)&&(n%3));

%o memoA355688 = Map();

%o A355688(n) = if(1==n,1,my(v); if(mapisdefined(memoA355688,n,&v), v, v = -sumdiv(n,d,if(d<n,A354354(n/d)*A355688(d),0)); mapput(memoA355688,n,v); (v)));

%o (PARI) A355688(n) = (moebius(n)*A354354(n)); \\ _Antti Karttunen_, Dec 20 2022

%Y Cf. A007310, A008683, A354354.

%Y Cf. also A355689, A355690.

%K sign,mult

%O 1

%A _Antti Karttunen_, Jul 15 2022