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A054535 Square array giving Ramanujan sum T(n,k) = c_n(k), where c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k), read by antidiagonals (n >= 1, k >= 1). 7
1, -1, 1, -1, 1, 1, 0, -1, -1, 1, -1, -2, 2, 1, 1, 1, -1, 0, -1, -1, 1, -1, -1, -1, 2, -1, 1, 1, 0, -1, -2, -1, 0, 2, -1, 1, 0, 0, -1, -1, 4, -2, -1, 1, 1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1, -1, -1, -3, -4, -1, 2, -1, 2, 2, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1, -1, 2, -1, -1, 0, 0, 6, -1, -1, -2, -1, 1, 1, 1, -1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

Replace the first column in A077049 with any k-th column in A177121 to get a new array. Then the matrix inverse of the new array will have the k-th column of A054535 (this array) as its first column. - Mats Granvik, May 03 2010

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10011 (T(n,k) for n+k <= 142).

Wikipedia, Ramanujan's sum

FORMULA

T(n,k) = phi(n)*mobius(n/gcd(n, k))/phi(n/gcd(n, k)). - Emeric Deutsch, Dec 23 2004

Dirichlet series: Sum_{n>=1} c_n(k)/n^s = sigma_{1-s}(k)/zeta(s) where sigma is the sum-of-divisors function. Sum_{n>=1} c_k(n)/n^s = zeta(s)*Sum_{d|k} mu(k/d)*d^(1-s). [Hardy & Wright, Titchmarsh] - R. J. Mathar, Apr 01 2012

From Mats Granvik, Oct 10 2016: (Start)

Let: n >= 1, k >= 1:

A(n,k) = if n mod k = 0 then k^r, otherwise 0;

B(n,k) = if n mod k = 0 then k/n^s, otherwise 0.

Then the Ramanujan's sum matrix equals:

inverse(A).transpose(B) evaluated at s=0 and r=0.

Equals inverse(A051731).transpose(A127093).

Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} T(n,k)/(n^r*k^s) = zeta(s)*zeta(s + r - 1)/zeta(r) as in Wikipedia. (End)

EXAMPLE

Square array starts:

  1, -1, -1,  0, -1, ...

  1,  1, -1, -2, -1, ...

  1, -1,  2,  0, -1, ...

  1,  1, -1,  2, -1, ...

  1, -1, -1,  0,  4, ...

  ...

MAPLE

with(numtheory): c:=(n, k)->phi(n)*mobius(n/gcd(n, k))/phi(n/gcd(n, k)): for n from 1 to 13 do seq(c(n+1-j, j), j=1..n) od; # gives the sequence in triangular form # Emeric Deutsch

MATHEMATICA

nmax = 14; t[n_, k_] := EulerPhi[n]*(MoebiusMu[n / GCD[n, k]] / EulerPhi[n / GCD[n, k]]); Flatten[ Table[t[n - k + 1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-Fran├žois Alcover, Nov 10 2011, after Emeric Deutsch *)

CROSSREFS

Transpose of array in A054534. Cf. A054532, A054533.

Cf. A086831=c_n(2), A085097=c_n(3), A085384=c_n(4), A085639=c_n(5), A085906=c_n(6), A099837=c_3(n), A176742=c_4(n), A100051=c_6(n).

Sequence in context: A111915 A066520 A088526 * A054534 A085769 A237422

Adjacent sequences:  A054532 A054533 A054534 * A054536 A054537 A054538

KEYWORD

sign,tabl,nice

AUTHOR

N. J. A. Sloane, Apr 09 2000

STATUS

approved

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Last modified May 24 04:25 EDT 2019. Contains 323528 sequences. (Running on oeis4.)