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A085906
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Ramanujan sum c_n(6).
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7
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1, 1, 2, -2, -1, 2, -1, 0, -3, -1, -1, -4, -1, -1, -2, 0, -1, -3, -1, 2, -2, -1, -1, 0, 0, -1, 0, 2, -1, -2, -1, 0, -2, -1, 1, 6, -1, -1, -2, 0, -1, -2, -1, 2, 3, -1, -1, 0, 0, 0, -2, 2, -1, 0, 1, 0, -2, -1, -1, 4, -1, -1, 3, 0, 1, -2, -1, 2, -2, 1, -1, 0, -1, -1, 0, 2, 1, -2, -1, 0, 0, -1, -1, 4, 1, -1, -2, 0, -1, 3, 1, 2, -2, -1, 1, 0, -1, 0, 3, 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.
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LINKS
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FORMULA
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a(n) = phi(n)*mu(n/gcd(n, 6)) / phi(n/gcd(n, 6)).
Dirichlet g.f. (1+2^(1-s)+3^(1-s)+6^(1-s))/zeta(s). - R. J. Mathar, Mar 26 2011
Lambert series and a consequence: Sum_{n >= 1} c_n(6) * z^n / (1 - z^n) = Sum_{s|6} s * z^s and -Sum_{n >= 1} (c_n(6) / n) * log(1 - z^n) = Sum_{s|6} z^s for |z| < 1 (using the principal value of the logarithm). - Petros Hadjicostas, Aug 24 2019
Multiplicative with a(2) = 1, a(2^2) = -2, and a(2^e) = 0 for e >= 3, a(3) = 2, a(3^2) = -3, and a(3^e) = 0 for e >= 3, and for a prime p >= 5, a(p) = -1, and a(p^e) = 0 for e >= 2.
Sum_{k=1..n} abs(a(k)) ~ (12/Pi^2) * n. (End)
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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[6/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[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := Switch[e, 1, 1, 2, -2, _, 0]; f[3, e_] := Switch[e, 1, 2, 2, -3, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 21 2024 *)
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PROG
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(PARI) a(n)=eulerphi(n)*moebius(n/gcd(n, 6))/eulerphi(n/gcd(n, 6))
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003
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EXTENSIONS
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STATUS
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approved
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