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%I #61 Sep 07 2019 07:04:52
%S 1,1,1,1,-1,2,1,1,-1,2,1,-1,-1,0,4,1,1,2,-2,-1,2,1,-1,-1,0,-1,1,6,1,1,
%T -1,2,-1,-1,-1,4,1,-1,2,0,-1,-2,-1,0,6,1,1,-1,-2,4,-1,-1,0,0,4,1,-1,
%U -1,0,-1,1,-1,0,0,1,10,1,1,2,2,-1,2,-1,-4,-3,-1,-1,4,1,-1,-1,0,-1,1,-1,0,0,1,-1,0,12,1
%N Ramanujan sum T(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2*Pi*i*m*n / k), triangular array read by rows for n >= 1 and 1 <= k <= n.
%C T(n, k) = c_k(n) = sum of the n-th powers of the k-th primitive roots of unity. - _Petros Hadjicostas_, Jul 27 2019
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160.
%H T. D. Noe, <a href="/A054532/b054532.txt">Rows n=1..50 of triangle, flattened</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 H. G. Gadiyar and R. Padma, <a href="https://arxiv.org/abs/math/0601574">Linking the circle and the sieve: Ramanujan-Fourier series</a>, arXiv:math/0601574 [math.NT], 2006.
%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 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 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 P. Moree and H. Hommerson, <a href="https://arxiv.org/abs/math/0307352">Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients</a>, arXiv:math/0307352 [math.NT], 2003.
%H K. Motose, <a href="http://escholarship.lib.okayama-u.ac.jp/mjou/vol47/iss1/5">Ramanujan's sums and cyclotomic polynomials</a>, Math. J. Okayama U. 47, no 1, (2005), Article 5.
%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 R. D. von Sterneck, <a href="https://play.google.com/books/reader?id=V1I-AQAAMAAJ&hl=de&printsec=frontcover&pg=GBS.PA1567">Ein Analogon zur additiven Zahlentheorie</a>, Sitzungsber. Akad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It may not be universally accessible.]
%H R. D. von Sterneck, <a href="https://eudml.org/doc/144877">Über ein Analogon zur additiven Zahlentheorie</a>, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113. [Summary of the 1902 paper.]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan%27s_sum">Ramanujan's sum</a>.
%H Wikipedia, <a href="https://de.wikipedia.org/wiki/Robert_Daublebsky_von_Sterneck_der_J%C3%BCngere">Robert Daublebsky von Sterneck der Jüngere</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 T(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} cos(2*Pi*m*n / k) = mu(k/gcd(k,n)) * phi(k) / phi(k/gcd(k,n)) = Sum_{d | gcd(k,n)} mu(k/d) * d. (All formulas were proved by Kluyver (1906, p. 410).) - _Petros Hadjicostas_, Aug 20 2019
%e Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
%e 1;
%e 1, 1;
%e 1, -1, 2;
%e 1, 1, -1, 2;
%e 1, -1, -1, 0, 4;
%e 1, 1, 2, -2, -1, 2;
%e 1, -1, -1, 0, -1, 1, 6;
%e 1, 1, -1, 2, -1, -1, -1, 4;
%e 1, -1, 2, 0, -1, -2, -1, 0, 6;
%e ...
%t t[n_, k_] := Sum[ c = Exp[2*Pi*I*m*(n/k)]; If[ GCD[m, k] == 1, c, 0], {m, 1, k}] // FullSimplify; Flatten[ Table[ t[n, k], {n, 1, 15}, {k, 1, n}]] (* _Jean-François Alcover_, Mar 15 2012 *)
%t (* to get the triangle in the example *)
%t TableForm[Table[t[n, k], {n, 1, 9}, {k, 1, n}]]
%t (* _Petros Hadjicostas_, Jul 27 2019 *)
%Y Cf. A054533, A054534, A054535, A282634.
%K sign,easy,nice,tabl
%O 1,6
%A _N. J. A. Sloane_, Apr 09 2000
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