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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-Fran├žois 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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Last modified October 20 15:14 EDT 2017. Contains 293612 sequences.