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%I #44 Jan 21 2024 02:22:01
%S 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,
%T 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,
%U 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
%N Ramanujan sum c_n(2).
%C Mobius transform of 1,2,0,0,0,0,... (A130779). - _R. J. Mathar_, Mar 24 2012
%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 edn., 1986.
%H Antti Karttunen, <a href="/A086831/b086831.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 Michael 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 Charles 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 Charles 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 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).
%F a(n) = phi(n)*mu(n/gcd(n, 2)) / phi(n/gcd(n, 2)).
%F Dirichlet g.f.: (1+2^(1-s))/zeta(s). [Titchmarsh eq. (1.5.4)] - _R. J. Mathar_, Mar 26 2011
%F 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
%F Sum_{k=1..n} abs(a(k)) ~ (8/Pi^2) * n. - _Amiram Eldar_, Jan 21 2024
%e 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
%p 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
%t 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 *)
%t 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 *)
%o (PARI) A086831(n) = (eulerphi(n)*moebius(n/gcd(n, 2))/eulerphi(n/gcd(n, 2))); \\ _Antti Karttunen_, Sep 27 2018
%Y Cf. A000010, A008683, A054532, A054533, A054534, A054535.
%Y Cf. A085097, A085384, A085639, A085906 for Ramanujan sums c_n(3), c_n(4), c_n(5), c_n(6).
%K sign,easy,mult
%O 1,4
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 07 2003
%E Corrected and extended by _Emeric Deutsch_, Dec 23 2004
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