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A054533 Triangular array giving Ramanujan sum T(n,k) = c_n(k), for n >= 1, 1<=k<=n, where c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k). 17
1, -1, 1, -1, -1, 2, 0, -2, 0, 2, -1, -1, -1, -1, 4, 1, -1, -2, -1, 1, 2, -1, -1, -1, -1, -1, -1, 6, 0, 0, 0, -4, 0, 0, 0, 4, 0, 0, -3, 0, 0, -3, 0, 0, 6, 1, -1, 1, -1, -4, -1, 1, -1, 1, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10, 0, 2, 0, -2, 0, -4, 0, -2, 0, 2, 0, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 12, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

From Wolfdieter Lang, Jan 06 2017: (Start)

Periodicity: c_n(k+n) = c_n(k). See the Apostol reference p. 161.

Multiplicativity: c_n(k)*c_m(k) = c_{n*m}(k), if gcd(n,m) = 1. For the proof see the Hardy reference, p. 138.

Dirichlet g.f. for fixed k: D(n,s) := Sum_{n>=1} c_n(k)/n^s = sigma_{1-s}(k)/zeta(s) = sigma_{s-1}(k)/(k^(s-1)*zeta(s)) for s > 1, with sigma_m(k) the sum of the m-th power of the divisors of k. See the Hardy reference, eqs. (9.6.1) and (9.6.2), pp. 139-140, or Hardy-Wright, Theorem 292, p. 250.

Sum_{n>=1} c_n(k)/n = 0. See the Hardy reference, p. 141. (End)

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 160-161.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 137-139.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Oxford Science Publications, Clarendon Press, Oxford, 2003, pp. 237-238.

LINKS

T. D. Noe, Rows n=1..50 of triangle, flattened

FORMULA

T(n, k) = Sum_{m=1..n, gcd(m,n) = 1} exp(2* Pi*I*m*k / n), n >= 1, 1 <= k <= n.

T(n, k) = Sum_{d | gcd(n,k)} d*Moebius(n/d), n >= 1, 1 <= k <= n.

EXAMPLE

Triangle begins

   1;

  -1,  1;

  -1, -1,  2;

   0, -2,  0,  2;

  -1, -1, -1, -1,  4;

   1, -1, -2, -1,  1,  2;

  -1, -1, -1, -1, -1, -1,  6;

   0,  0,  0, -4,  0,  0,  0,  4;

   0,  0, -3,  0,  0, -3,  0,  0,  6;

   1, -1,  1, -1, -4, -1,  1, -1,  1,  4;

  -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10;

   0,  2,  0, -2,  0, -4,  0, -2,  0,  2,  0,  4;

  -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 12;

[Edited by Jon E. Schoenfield, Jan 03 2017]

Periodicity and multiplicativity: c_6(k) = c_2(k)*c_3(k), e.g.: 2 = c_6(6) = c_2(6)*c_3(6) = c_2(2)*c_3(3) = 1*2 = 2. - Wolfdieter Lang, Jan 05 2017

MATHEMATICA

c[k_, n_] := Sum[ If[GCD[m, k] == 1, Exp[2 Pi*I*m*n/k], 0], {m, 1, k}]; A054533 = Flatten[ Table[c[n, k] // FullSimplify, {n, 1, 14}, {k, 1, n}] ] (* Jean-François Alcover, Jun 27 2012 *)

CROSSREFS

Cf. A008683, A054532, A054534, A054535.

Sequence in context: A117199 A230632 A052511 * A227957 A247977 A143232

Adjacent sequences:  A054530 A054531 A054532 * A054534 A054535 A054536

KEYWORD

sign,easy,nice,tabl

AUTHOR

N. J. A. Sloane, Apr 09 2000

STATUS

approved

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Last modified May 26 07:30 EDT 2017. Contains 287092 sequences.