A085906 Ramanujan sum c_n(6).
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, 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, 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
Offset: 1
References
- Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
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.
- Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
- Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
- J. C. Kluyver, Some formulae concerning the integers less than n and prime to n, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414; see p. 410.
- 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.
- M. V. Subbarao, The Brauer-Rademacher identity, Amer. Math. Monthly 72 (1965), 135-138.
- Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
- Wikipedia, Ramanujan's sum.
- Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.
Crossrefs
Programs
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Mathematica
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 *) 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 *) -
PARI
a(n)=eulerphi(n)*moebius(n/gcd(n,6))/eulerphi(n/gcd(n,6))
Formula
a(n) = phi(n)*mu(n/gcd(n, 6)) / phi(n/gcd(n, 6)).
Dirichlet g.f. (1+2^(1-s)+3^(1-s)+6^(1-s))/zeta(s). - R. J. Mathar, Mar 26 2011
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
From Amiram Eldar, Jan 21 2024: (Start)
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.
Sum_{k=1..n} abs(a(k)) ~ (12/Pi^2) * n. (End)
Extensions
More terms from Benoit Cloitre, Aug 18 2003