A086831 Ramanujan sum c_n(2).

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

Offset: 1

Author

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

Keywords

• sign
• easy
• mult

Comments

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

Examples

			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
		

References

• 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 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.
• Michael 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.
• Charles 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.
• Charles 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.

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).

Programs

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 *)
f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := Switch[e, 1, 1, 2, -2, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 21 2024 *)

PARI

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

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

Multiplicative with a(2) = 1, a(2^2) = -2, and a(2^e) = 0 for e >= 3, and for an odd prime p, a(p) = -1 and a(p^e) = 0 for e >= 2. - Amiram Eldar, Sep 14 2023

Sum_{k=1..n} abs(a(k)) ~ (8/Pi^2) * n. - Amiram Eldar, Jan 21 2024

Extensions

Corrected and extended by Emeric Deutsch, Dec 23 2004