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Dirichlet inverse of A359377, where A359377(n) = 1 if 3*n is squarefree, otherwise 0.
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%I #20 Jan 03 2023 09:21:28

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

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

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

%N Dirichlet inverse of A359377, where A359377(n) = 1 if 3*n is squarefree, otherwise 0.

%C Note the correspondences between four sequences:

%C A156277 --- abs ---> A359377

%C ^ ^

%C | |

%C inv inv

%C | |

%C v v

%C A011655 <--- abs --- A359378 (this sequence)

%C Here inv means that the sequences are Dirichlet Inverses of each other, and abs means taking absolute values.

%H Antti Karttunen, <a href="/A359378/b359378.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} A359377(n/d) * a(d).

%F Multiplicative with a(3^e) = 0 and a(p^k) = (-1)^k for all primes p <> 3.

%F a(n) = A359170(n) - A359172(n).

%F For all n >= 1, a(A001651(n)) = A008836(A001651(n)).

%F Dirichlet g.f.: 3^s/((3^s-1)*zeta(s)). - _Amiram Eldar_, Jan 03 2023

%t f[p_, e_] := (-1)^e; f[3, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2022 *)

%o (PARI) A359378(n) = { my(f = factor(n)); prod(k=1, #f~, (3!=f[k,1])*((-1)^f[k, 2])); };

%Y Cf. A001651, A008836, A011655 (absolute values), A156277 (Dirichlet inverse of the absolute values), A359377 (Dirichlet inverse).

%Y Cf. A008585 (after its initial term gives the positions of 0's), A359171 (of positive terms), A359381 (of negative terms), A359170, A359172.

%Y Cf. also A166698, A358839.

%K sign,mult

%O 1

%A _Antti Karttunen_, Dec 29 2022