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A085097
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Ramanujan sum c_n(3).
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7
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1, -1, 2, 0, -1, -2, -1, 0, -3, 1, -1, 0, -1, 1, -2, 0, -1, 3, -1, 0, -2, 1, -1, 0, 0, 1, 0, 0, -1, 2, -1, 0, -2, 1, 1, 0, -1, 1, -2, 0, -1, 2, -1, 0, 3, 1, -1, 0, 0, 0, -2, 0, -1, 0, 1, 0, -2, 1, -1, 0, -1, 1, 3, 0, 1, 2, -1, 0, -2, -1, -1, 0, -1, 1, 0, 0, 1, 2, -1, 0, 0, 1, -1, 0, 1, 1, -2, 0, -1, -3, 1, 0, -2, 1, 1, 0, -1, 0, 3, 0, -1, 2, -1, 0, 2, 1, -1, 0
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OFFSET
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1,3
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REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Acad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It seems that his father, Robert Freiherr Daublebsky von Sterneck, had exactly the same name.]
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LINKS
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FORMULA
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a(n) = phi(n)*mu(n/gcd(n, 3)) / phi(n/gcd(n, 3)).
Dirichlet g.f.: (1+3^(1-s))/zeta(s). [Titchmarsh eq. (1.5.4.)] - R. J. Mathar, Mar 26 2011
Multiplicative with a(3) = 2, a(3^2) = -3, a(3^e) = 0 for e >= 3, for a prime p != 3, a(p) = -1 and a(p^e) = 0 for e >= 2. - Amiram Eldar, Sep 10 2023
Sum_{k=1..n} abs(a(k)) ~ (9/Pi^2) * n. - Amiram Eldar, Jan 21 2024
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MATHEMATICA
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f[list_, i_] := list[[i]]; nn = 105; a =Table[MoebiusMu[n], {n, 1, nn}]; b =Table[If[IntegerQ[3/n], n, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Dec 30 2015 *)
f[3, e_] := Switch[e, 1, 2, 2, -3, _, 0]; f[p_, e_] := If[e == 1, -1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2023 *)
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PROG
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(PARI) a(n)=eulerphi(n)*moebius(n/gcd(n, 3))/eulerphi(n/gcd(n, 3))
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 10 2003
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EXTENSIONS
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STATUS
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approved
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