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 A054532 Triangular array giving Ramanujan sum T(n,k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k), n >= 1, 1<=k<=n. 5
 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, -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, -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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 REFERENCES T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160. LINKS T. D. Noe, Rows n=1..50 of triangle, flattened P. Moree and H. Hommerson, Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients H. G. Gadiyar and R. Padma, Linking the circle and the sieve: Ramanujan-Fourier series K. Motose, Ramanujan's sums and cyclotomic polynomials, Math. J. Okayama U. 47, no 1, (2005), Article 5. EXAMPLE 1; 1,1; 1,-1,2; 1,1,-1,2; 1,-1,-1,0,4; 1,1,2,-2,-1,2; ... MATHEMATICA 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 *) CROSSREFS Cf. A054533, A054534, A054535. Sequence in context: A105242 A221362 A114116 * A260415 A214710 A120888 Adjacent sequences:  A054529 A054530 A054531 * A054533 A054534 A054535 KEYWORD sign,easy,nice,tabl AUTHOR N. J. A. Sloane, Apr 09 2000 STATUS approved

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