%I #39 Jan 21 2024 02:20:56

%S 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,

%T -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,

%U 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

%N Ramanujan sum c_n(3).

%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.

%D E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.

%D 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.]

%H Antti Karttunen, <a href="/A085097/b085097.txt">Table of n, a(n) for n = 1..65537</a>

%H Tom M. Apostol, <a href="https://projecteuclid.org/download/pdf_1/euclid.pjm/1102968273">Arithmetical properties of generalized Ramanujan sums</a>, Pacific J. Math. 41 (1972), 281-293.

%H Eckford Cohen, <a href="https://dx.doi.org/10.1073/pnas.41.11.939">A class of arithmetic functions</a>, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.

%H A. Elashvili, M. Jibladze, and D. Pataraia, <a href="http://dx.doi.org/10.1023/A:1018727630642">Combinatorics of necklaces and "Hermite reciprocity"</a>, J. Algebraic Combin. 10 (1999), 173-188.

%H M. L. Fredman, <a href="https://doi.org/10.1016/0097-3165(75)90008-4">A symmetry relationship for a class of partitions</a>, J. Combinatorial Theory Ser. A 18 (1975), 199-202.

%H Otto Hölder, <a href="http://matwbn.icm.edu.pl/ksiazki/pmf/pmf43/pmf4312.pdf">Zur Theorie der Kreisteilungsgleichung K_m(x)=0</a>, Prace mat.-fiz. 43 (1936), 13-23.

%H C. A. Nicol, <a href="https://dx.doi.org/10.1073/pnas.39.9.963">On restricted partitions and a generalization of the Euler phi number and the Moebius function</a>, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.

%H C. A. Nicol and H. S. Vandiver, <a href="https://dx.doi.org/10.1073/pnas.40.9.825 ">A von Sterneck arithmetical function and restricted partitions with respect to a modulus</a>, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.

%H K. G. Ramanathan, <a href="https://www.ias.ac.in/article/fulltext/seca/020/01/0062-0069">Some applications of Ramanujan's trigonometrical sum C_m(n)</a>, Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.

%H Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram21.pdf">On certain trigonometric sums and their applications in the theory of numbers</a>, Trans. Camb. Phil. Soc. 22 (1918), 259-276.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan%27s_sum">Ramanujan's sum</a>.

%H Aurel Wintner, <a href="https://www.jstor.org/stable/2371672">On a statistics of the Ramanujan sums</a>, Amer. J. Math., 64(1) (1942), 106-114.

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

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

%F 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

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

%t 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 *)

%t 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 *)

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

%Y Cf. A000010, A008683, A086831, A085906.

%K sign,easy,mult

%O 1,3

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

%E More terms from _Benoit Cloitre_, Aug 12 2003