A085639 Ramanujan sum c_n(5).

1, -1, -1, 0, 4, 1, -1, 0, 0, -4, -1, 0, -1, 1, -4, 0, -1, 0, -1, 0, 1, 1, -1, 0, -5, 1, 0, 0, -1, 4, -1, 0, 1, 1, -4, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 5, 1, 0, -1, 0, -4, 0, 1, 1, -1, 0, -1, 1, 0, 0, -4, -1, -1, 0, 1, 4, -1, 0, -1, 1, 5, 0, 1, -1, -1, 0, 0, 1, -1, 0, -4, 1, 1, 0, -1

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 15 2003

Keywords

• sign
• mult

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

Cf. A000010, A008683, A054532, A054533, A054534, A054535, A085384, A085906, A085097, A086831.

Programs

Mathematica

a[n_] := EulerPhi[n] * MoebiusMu[n/GCD[n, 5]] / EulerPhi[n/GCD[n, 5]]; Table[ a[n], {n, 1, 105}]
f[p_, e_] := If[e == 1, -1, 0]; f[5, e_] := Switch[e, 1, 4, 2, -5, , 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,5))/eulerphi(n/gcd(n,5))

Formula

a(n) = phi(n)*mu(n/gcd(n, 5)) / phi(n/gcd(n, 5)).

Dirichlet g.f.: (1+5^(1-s))/zeta(s). - R. J. Mathar, Mar 26 2011

Lambert series and a consequence: Sum_{n >= 1} c_n(5) * z^n / (1 - z^n) = z + 5*z^5 and -Sum_{n >= 1} (c_n(5) / n) * log(1 - z^n) = z + z^5 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(5) = 4, a(5^2) = -5, and a(5^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)) ~ (10/Pi^2) * n. (End)

Extensions

More terms from Robert G. Wilson v and Benoit Cloitre, Aug 17 2003