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%I #32 Jan 21 2024 02:21:45
%S 1,1,2,-2,-1,2,-1,0,-3,-1,-1,-4,-1,-1,-2,0,-1,-3,-1,2,-2,-1,-1,0,0,-1,
%T 0,2,-1,-2,-1,0,-2,-1,1,6,-1,-1,-2,0,-1,-2,-1,2,3,-1,-1,0,0,0,-2,2,-1,
%U 0,1,0,-2,-1,-1,4,-1,-1,3,0,1,-2,-1,2,-2,1,-1,0,-1,-1,0,2,1,-2,-1,0,0,-1,-1,4,1,-1,-2,0,-1,3,1,2,-2,-1,1,0,-1,0,3,0
%N Ramanujan sum c_n(6).
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
%H Antti Karttunen, <a href="/A085906/b085906.txt">Table of n, a(n) for n = 1..65537</a>
%H Tom M. Apostol, <a href="https://projecteuclid.org/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 Emiliano Gagliardo, <a href="http://www.bdim.eu/item?id=BUMI_1953_3_8_3_269_0">Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità</a>, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
%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 J. C. Kluyver, <a href="https://www.dwc.knaw.nl/DL/publications/PU00013765.pdf">Some formulae concerning the integers less than n and prime to n</a>, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414; see p. 410.
%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 M. V. Subbarao, <a href="https://www.jstor.org/stable/2310974">The Brauer-Rademacher identity</a>, Amer. Math. Monthly 72 (1965), 135-138.
%H Peter H. van der Kamp, <a href="http://emis.impa.br/EMIS/journals/INTEGERS/papers/n24/n24.Abstract.html">On the Fourier transform of the greatest common divisor</a>, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
%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, 6)) / phi(n/gcd(n, 6)).
%F Dirichlet g.f. (1+2^(1-s)+3^(1-s)+6^(1-s))/zeta(s). - _R. J. Mathar_, Mar 26 2011
%F Lambert series and a consequence: Sum_{n >= 1} c_n(6) * z^n / (1 - z^n) = Sum_{s|6} s * z^s and -Sum_{n >= 1} (c_n(6) / n) * log(1 - z^n) = Sum_{s|6} z^s for |z| < 1 (using the principal value of the logarithm). - _Petros Hadjicostas_, Aug 24 2019
%F From _Amiram Eldar_, Jan 21 2024: (Start)
%F Multiplicative with a(2) = 1, a(2^2) = -2, and a(2^e) = 0 for e >= 3, a(3) = 2, a(3^2) = -3, and a(3^e) = 0 for e >= 3, and for a prime p >= 5, a(p) = -1, and a(p^e) = 0 for e >= 2.
%F Sum_{k=1..n} abs(a(k)) ~ (12/Pi^2) * n. (End)
%t f[list_, i_] := list[[i]]; nn = 105; a =Table[MoebiusMu[n], {n, 1, nn}]; b =Table[If[IntegerQ[6/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]; f[3, e_] := Switch[e, 1, 2, 2, -3, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 21 2024 *)
%o (PARI) a(n)=eulerphi(n)*moebius(n/gcd(n,6))/eulerphi(n/gcd(n,6))
%Y Cf. A086831, A085097, A085384, A085639 for Ramanujan sums c_n(2) .. c_n(5).
%Y Cf. A000010, A008683.
%K sign,mult
%O 1,3
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003
%E More terms from _Benoit Cloitre_, Aug 18 2003
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