A085097 Ramanujan sum c_n(3).

1, -1, 2, 0, -1, -2, -1, 0, -3, 1, -1, 0, -1, 1, -2, 0, -1, 3, -1, 0, -2, 1, -1, 0, 0, 1, 0, 0, -1, 2, -1, 0, -2, 1, 1, 0, -1, 1, -2, 0, -1, 2, -1, 0, 3, 1, -1, 0, 0, 0, -2, 0, -1, 0, 1, 0, -2, 1, -1, 0, -1, 1, 3, 0, 1, 2, -1, 0, -2, -1, -1, 0, -1, 1, 0, 0, 1, 2, -1, 0, 0, 1, -1, 0, 1, 1, -2, 0, -1, -3, 1, 0, -2, 1, 1, 0, -1, 0, 3, 0, -1, 2, -1, 0, 2, 1, -1, 0

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 10 2003

Keywords

• sign
• easy
• mult

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 ed., 1986.
• R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Acad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It seems that his father, Robert Freiherr Daublebsky von Sterneck, had exactly the same name.]

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.

Crossrefs

Cf. A000010, A008683, A086831, A085906.

Programs

Mathematica

f[list_, i_] := list[[i]]; nn = 105; a =Table[MoebiusMu[n], {n, 1, nn}]; b =Table[If[IntegerQ[3/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[3, e_] := Switch[e, 1, 2, 2, -3, , 0]; f[p, e_] := If[e == 1, -1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2023 *)

PARI

a(n)=eulerphi(n)*moebius(n/gcd(n,3))/eulerphi(n/gcd(n,3))

Formula

a(n) = phi(n)*mu(n/gcd(n, 3)) / phi(n/gcd(n, 3)).

Dirichlet g.f.: (1+3^(1-s))/zeta(s). [Titchmarsh eq. (1.5.4.)] - R. J. Mathar, Mar 26 2011

Multiplicative with a(3) = 2, a(3^2) = -3, a(3^e) = 0 for e >= 3, for a prime p != 3, a(p) = -1 and a(p^e) = 0 for e >= 2. - Amiram Eldar, Sep 10 2023

Sum_{k=1..n} abs(a(k)) ~ (9/Pi^2) * n. - Amiram Eldar, Jan 21 2024

Extensions

More terms from Benoit Cloitre, Aug 12 2003