A085639 - OEIS

%I #35 Jan 21 2024 02:21:26

%S 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,

%T -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,

%U 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

%N Ramanujan sum c_n(5).

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

%H Antti Karttunen, <a href="/A085639/b085639.txt">Table of n, a(n) for n = 1..65537</a>

%H Tom M. Apostol, <a href="http://dx.doi.org/10.2140/pjm.1972.41.281">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, 5)) / phi(n/gcd(n, 5)).

%F Dirichlet g.f.: (1+5^(1-s))/zeta(s). - _R. J. Mathar_, Mar 26 2011

%F 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

%F From _Amiram Eldar_, Jan 21 2024: (Start)

%F 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.

%F Sum_{k=1..n} abs(a(k)) ~ (10/Pi^2) * n. (End)

%t a[n_] := EulerPhi[n] * MoebiusMu[n/GCD[n, 5]] / EulerPhi[n/GCD[n, 5]]; Table[ a[n], {n, 1, 105}]

%t 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 *)

%o (PARI) a(n)=eulerphi(n)*moebius(n/gcd(n,5))/eulerphi(n/gcd(n,5))

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

%K sign,mult

%O 1,5

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

%E More terms from _Robert G. Wilson v_ and _Benoit Cloitre_, Aug 17 2003