This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A085384 #43 Jan 21 2024 02:21:14
%S A085384 1,1,-1,2,-1,-1,-1,-4,0,-1,-1,-2,-1,-1,1,0,-1,0,-1,-2,1,-1,-1,4,0,-1,
%T A085384 0,-2,-1,1,-1,0,1,-1,1,0,-1,-1,1,4,-1,1,-1,-2,0,-1,-1,0,0,0,1,-2,-1,0,
%U A085384 1,4,1,-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
%N A085384 Ramanujan sum c_n(4).
%D A085384 Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
%D A085384 E. C. Titchmarsh and D. R. Heath-Brown, The Theory of the Riemann Zeta-function, 2nd ed., 1986.
%D A085384 R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Acad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa).
%H A085384 Antti Karttunen, <a href="/A085384/b085384.txt">Table of n, a(n) for n = 1..65537</a>
%H A085384 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 A085384 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 A085384 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 A085384 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 A085384 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 A085384 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 A085384 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 A085384 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 A085384 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 A085384 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 A085384 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan%27s_sum">Ramanujan's sum</a>.
%H A085384 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 A085384 a(n) = phi(n)*mu(n/gcd(n, 4)) / phi(n/gcd(n, 4)).
%F A085384 Dirichlet g.f.: (1+2^(1-s)+4^(1-s))/zeta(s). [Titchmarsh] - _R. J. Mathar_, Mar 26 2011
%F A085384 Lambert series and a consequence: Sum_{n >= 1} c_n(4) * z^n / (1 - z^n) = Sum_{s|4} s * z^s and -Sum_{n >= 1} (c_n(4) / n) * log(1 - z^n) = Sum_{s|4} z^s for |z| < 1 (using the principal value of the logarithm). - _Petros Hadjicostas_, Aug 24 2019
%F A085384 From _Amiram Eldar_, Jan 21 2024: (Start)
%F A085384 Multiplicative with a(2) = 1, a(2^2) = 2, a(2^3) = -4, and a(2^e) = 0 for e >= 4, and for an odd prime p, a(p) = -1, and a(p^e) = 0 for e >= 2.
%F A085384 Sum_{k=1..n} abs(a(k)) ~ (10/Pi^2) * n. (End)
%t A085384 a[n_] := EulerPhi[n] * MoebiusMu[n/GCD[n, 4]] / EulerPhi[n/GCD[n, 4]]; Table[ a[n], {n, 1, 105}]
%t A085384 f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := Switch[e, 1, 1, 2, 2, 3, -4, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 21 2024 *)
%o A085384 (PARI) a(n)=eulerphi(n)*moebius(n/gcd(n,4))/eulerphi(n/gcd(n,4))
%Y A085384 Cf. A000010, A008683, A054532, A054533, A054534, A054535, A085097, A086831, A085639.
%K A085384 sign,mult
%O A085384 1,4
%A A085384 Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 12 2003
%E A085384 More terms from _Robert G. Wilson v_ and _Benoit Cloitre_, Aug 17 2003