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A085384 Ramanujan sum c_n(4). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

REFERENCES

T. 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).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.

Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.

A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.

M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.

Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.

J. C. Kluyver, Some formulae concerning the integers less than n and prime to n, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414; see p. 410.

C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.

C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.

K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.

Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.

Wikipedia, Ramanujan's sum.

Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

FORMULA

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

MATHEMATICA

a[n_] := EulerPhi[n] * MoebiusMu[n/GCD[n, 4]] / EulerPhi[n/GCD[n, 4]]; Table[ a[n], {n, 1, 105}]

PROG

(PARI) a(n)=eulerphi(n)*moebius(n/gcd(n, 4))/eulerphi(n/gcd(n, 4))

CROSSREFS

Cf. A054532, A054533, A054534, A054535, A085097, A086831, A085639.

Sequence in context: A007442 A054772 A294616 * A067856 A160467 A122374

Adjacent sequences:  A085381 A085382 A085383 * A085385 A085386 A085387

KEYWORD

sign,mult

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 12 2003

EXTENSIONS

More terms from Robert G. Wilson v and Benoit Cloitre, Aug 17 2003

STATUS

approved

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Last modified June 4 04:39 EDT 2020. Contains 334815 sequences. (Running on oeis4.)