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 A054534 Square array giving Ramanujan sum T(n,k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k), read by antidiagonals upwards (n >= 1, k >= 1). 5
 1, 1, -1, 1, 1, -1, 1, -1, -1, 0, 1, 1, 2, -2, -1, 1, -1, -1, 0, -1, 1, 1, 1, -1, 2, -1, -1, -1, 1, -1, 2, 0, -1, -2, -1, 0, 1, 1, -1, -2, 4, -1, -1, 0, 0, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, 1, 1, 2, 2, -1, 2, -1, -4, -3, -1, -1, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 1, 1, -1, -2, -1, -1, 6, 0, 0, -1, -1, 2, -1, 1, -1, 2, 0, 4, -2, -1, 0, -3, -4, -1, 0, -1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,13 REFERENCES T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160. H. Rademacher, Collected Papers of Hans Rademacher, vol. II, MIT Press, 1974, p. 435. S. Ramanujan, On Certain Trigonometrical Sums and their Applications in the Theory of Numbers, pp. 179-199 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea Publishing 2000. LINKS Seiichi Manyama, Antidiagonals n = 1..140, flattened Wikipedia, Ramanujan's sum FORMULA T(n, 1) = c_1(n) = 1. T(n, 2) = c_2(n) = A033999(n). T(n, 3) = c_3(n) = A099837(n) if n>1. T(n, 4) = c_4(n) = A176742(n) if n>1. T(n, 6) = c_6(n) = A100051(n) if n>1. - Michael Somos, Mar 21 2011 T(1, n) = c_n(1) = A008683(n). T(2, n) = c_n(2) = A086831(n). T(3, n) = c_n(3) = A085097(n). T(4, n) = c_n(4) = A085384(n). T(5, n) = c_n(5) = A085639(n). T(6, n) = c_n(6) = A085906(n). - Michael Somos, Mar 21 2011 T(n, n) = T(k * n, n) = A000010(n), T(n, 2*n) = -A062570(n). - Michael Somos, Mar 21 2011 EXAMPLE Array begins 1, -1, -1,  0 ... 1,  1, -1, -2 ... 1, -1,  2,  0 ... 1,  1, -1,  2 ... 1, -1, -1,  0 ... MATHEMATICA nmax = 14; mu[n_Integer] = MoebiusMu[n]; mu[_] = 0; t[n_, k_] := Total[ #*mu[k/#]& /@ Divisors[n]]; Flatten[ Table[ t[n-k+1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 14 2011, after Pari *) PROG (PARI) {T(n, k) = if( n<1 || k<1, 0, sumdiv( n, d, if( k%d==0, d * moebius(k / d))))} /* Michael Somos, Dec 05 2002 */ (PARI) {T(n, k) = if( n<1 || k<1, 0, polsym( polcyclo( k), n) [n + 1])} /* Michael Somos, Mar 21 2011 */ CROSSREFS Cf. A000010, A033999, A054532, A054533, A054535, A062570, A085097, A058384, A085639, A085906, A099837, A100051, A176742. Sequence in context: A066520 A088526 A054535 * A085769 A237422 A102552 Adjacent sequences:  A054531 A054532 A054533 * A054535 A054536 A054537 KEYWORD sign,tabl,nice AUTHOR N. J. A. Sloane, Apr 09 2000 STATUS approved

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Last modified September 25 11:05 EDT 2018. Contains 315389 sequences. (Running on oeis4.)