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A358839
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Dirichlet inverse of A353627, the characteristic function of the squarefree numbers multiplied by binary powers.
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11
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1, -1, -1, 0, -1, 1, -1, 0, 1, 1, -1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 1, 1, -1, 0, 1, 1, -1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1, 1, 0, 1, 1, -1, 0, -1, 1, -1, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, 1, 1, 1, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1
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OFFSET
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1
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COMMENTS
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Note the correspondences between four sequences:
^ ^
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inv inv
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v v
Here inv means that the sequences are Dirichlet Inverses of each other, and abs means taking absolute values.
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (-1)^e for odd primes p, and a(2^e) = -1 if e = 1, otherwise 0.
For all e >= 0, a(2^e) = A008683(2^e).
For all n >= 0, a(2n+1) = A008836(2n+1).
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A353627(n/d) * a(d).
Dirichlet g.f.: (1-1/4^s)*zeta(2*s)/zeta(s). - Amiram Eldar, Jan 01 2023
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MATHEMATICA
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f[p_, e_] := (-1)^e; f[2, e_] := If[e == 1, -1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 28 2022 *)
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PROG
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(PARI) A358839(n) = { my(f = factor(n)); prod(k=1, #f~, if(2==f[k, 1], -(1==f[k, 2]), (-1)^f[k, 2])); };
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CROSSREFS
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Cf. A166486 (absolute values), A353627 (Dirichlet inverse), A355689 (Dirichlet inverse of the absolute values).
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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