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A085384
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Ramanujan sum c_n(4).
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6
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1, 1, -1, 2, -1, -1, -1, -4, 0, -1, -1, -2, -1, -1, 1, 0, -1, 0, -1, -2, 1, -1, -1, 4, 0, -1, 0, -2, -1, 1, -1, 0, 1, -1, 1, 0, -1, -1, 1, 4, -1, 1, -1, -2, 0, -1, -1, 0, 0, 0, 1, -2, -1, 0, 1, 4, 1, -1, -1, 2, -1, -1, 0, 0, 1, 1, -1, -2, 1, 1, -1, 0, -1, -1, 0, -2, 1, 1, -1, 0, 0, -1, -1, 2
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OFFSET
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1,4
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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).
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LINKS
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FORMULA
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a(n) = phi(n)*mu(n/gcd(n, 4)) / phi(n/gcd(n, 4)).
Dirichlet g.f.: (1+2^(1-s)+4^(1-s))/zeta(s). [Titchmarsh] - R. J. Mathar, Mar 26 2011
Lambert series and a consequence: Sum_{n >= 1} c_n(4) * z^n / (1 - z^n) = Sum_{s|4} s * z^s and -Sum_{n >= 1} (c_n(4) / n) * log(1 - z^n) = Sum_{s|4} 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, a(2^3) = -4, and a(2^e) = 0 for e >= 4, and for an odd prime p, a(p) = -1, and a(p^e) = 0 for e >= 2.
Sum_{k=1..n} abs(a(k)) ~ (10/Pi^2) * n. (End)
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MATHEMATICA
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a[n_] := EulerPhi[n] * MoebiusMu[n/GCD[n, 4]] / EulerPhi[n/GCD[n, 4]]; Table[ a[n], {n, 1, 105}]
f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := Switch[e, 1, 1, 2, 2, 3, -4, _, 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, 4))/eulerphi(n/gcd(n, 4))
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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 12 2003
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EXTENSIONS
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STATUS
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approved
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