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A086831 Ramanujan sum c_n(2). 7
1, 1, -1, -2, -1, -1, -1, 0, 0, -1, -1, 2, -1, -1, 1, 0, -1, 0, -1, 2, 1, -1, -1, 0, 0, -1, 0, 2, -1, 1, -1, 0, 1, -1, 1, 0, -1, -1, 1, 0, -1, 1, -1, 2, 0, -1, -1, 0, 0, 0, 1, 2, -1, 0, 1, 0, 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, 1, -1, 1, 0, -1, 0, 1, 2, 1, -1, 1, 0, -1, 0, 0, 0, -1, 1, -1, 0, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Mobius transform of 1,2,0,0,0,0,... (A130779). - R. J. Mathar, Mar 24 2012

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 edn., 1986.

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.

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

For a general k >= 1, c_n(k) = phi(n)*mu(n/gcd(n, k)) / phi(n/gcd(n, k)); so c_n(1) = mu(n) = A008683(n).

a(n) = phi(n)*mu(n/gcd(n, 2)) / phi(n/gcd(n, 2)).

Dirichlet g.f.: (1+2^(1-s))/zeta(s). [Titchmarsh eq. (1.5.4)] - R. J. Mathar, Mar 26 2011

EXAMPLE

a(4) = -2 because the primitive fourth roots of unity are i and -i.  We sum their squares to get i^2 + (-i)^2 = -1 + -1 = -2. - Geoffrey Critzer, Dec 30 2015

MAPLE

with(numtheory):a:=n->phi(n)*mobius(n/gcd(n, 2))/phi(n/gcd(n, 2)): seq(a(n), n=1..130); # Emeric Deutsch, Dec 23 2004

MATHEMATICA

f[list_, i_] := list[[i]]; nn = 105; a = Table[MoebiusMu[n], {n, 1, nn}]; b =Table[If[IntegerQ[2/n], n, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Dec 30 2015 *)

PROG

(PARI) A086831(n) = (eulerphi(n)*moebius(n/gcd(n, 2))/eulerphi(n/gcd(n, 2))); \\ Antti Karttunen, Sep 27 2018

CROSSREFS

Cf. A000010, A008683, A054532, A054533, A054534, A054535.

Cf. A085097, A085384, A085639, A085906 for Ramanujan sums c_n(3), c_n(4), c_n(5), c_n(6).

Sequence in context: A253638 A337586 A345006 * A191340 A211229 A335621

Adjacent sequences:  A086828 A086829 A086830 * A086832 A086833 A086834

KEYWORD

sign,easy,mult

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 07 2003

EXTENSIONS

Corrected and extended by Emeric Deutsch, Dec 23 2004

STATUS

approved

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Last modified June 23 09:43 EDT 2021. Contains 345397 sequences. (Running on oeis4.)