A282634 -id:A282634 - cp's OEIS frontend

A054533 Triangular array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) for n >= 1 and 1 <= k <= n.

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

Offset: 1

N. J. A. Sloane, Apr 09 2000

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)
Right border gives A000010. - Omar E. Pol, May 08 2018
Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} T(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = A051168(n + k, k). - Petros Hadjicostas, Jul 09 2019
We have T(n, k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) and A054532(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k) for n >= 1 and 1 <= k <= n. - Petros Hadjicostas, Jul 27 2019

			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

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.

Seiichi Manyama, Rows n=1..140 of triangle, flattened (Rows 1..50 from T. D. Noe)
Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
J. C. Kluyver, Some formulae concerning the integers less than n and prime to n, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414. [See bottom of p. 410, where the author proves that Sum cos(2*Pi*q*v/n) = mu(n/D) * phi(n) /phi(n/D), where D is the gcd of n and q. The summation is over integers v "less than n and prime to n" (top of p. 408).]
C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Akad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa).
R. D. von Sterneck, Über ein Analogon zur additiven Zahlentheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113.
M. V. Subbarao, The Brauer-Rademacher identity, Amer. Math. Monthly 72 (1965), 135-138.
Wikipedia, Ramanujan's sum.
Wikipedia, Robert Daublebsky von Sterneck der Jüngere.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Cf. A008683, A054532, A054534, A054535, A282634, A000010.

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 *)
(* to get the triangle in the example above *)
FormTable[Table[c[n, k] // FullSimplify, {n, 1, 13}, {k, 1, n}]]
(* Petros Hadjicostas, Jul 28 2019 *)

T(n,k) = sumdiv(gcd(n,k), d, d*moebius(n/d));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(T(n,k), ", "); ); print(); ); }; \\ Michel Marcus, Jun 14 2018

T(n, k) = Sum_{m=1..n, gcd(m,n) = 1} exp(2*Pi*i*m*k / n), n >= 1, 1 <= k <= n, where i is the imaginary unit.

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

Name edited by Petros Hadjicostas, Jul 27 2019

A054535 Square array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n), read by antidiagonals upwards (n >= 1, k >= 1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 09 2000

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
We have T(n, k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) and
A054534(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k). That is, the current array is the transpose of array A054534. Dirichlet g.f.'s for these two arrays are given below by R. J. Mathar and  Mats Granvik. - Petros Hadjicostas, Jul 27 2019

			Square array T(n,k) = c_n(k) (with rows n >= 1 and columns k >= 1) starts as follows:
   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  -1,  1, -1,  1, -1,  1, -1,  1, -1,  1, -1,  1, -1, ...
  -1, -1,  2, -1, -1,  2, -1, -1,  2, -1, -1,  2, -1, ...
   0, -2,  0,  2,  0, -2,  0,  2,  0, -2,  0,  2,  0, ...
  -1, -1, -1, -1,  4, -1, -1, -1, -1,  4, -1, -1, -1, ...
   1, -1, -2, -1,  1,  2,  1, -1, -2, -1,  1,  2,  1, ...
  -1, -1, -1, -1, -1, -1,  6, -1, -1, -1, -1, -1, -1, ...
   0,  0,  0, -4,  0,  0,  0,  4,  0,  0,  0, -4,  0, ...
   ... [example edited by _Petros Hadjicostas_, Jul 27 2019]

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Oxford Science Publications, Clarendon Press, Oxford, 2003.
E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.

Robert Israel, Table of n, a(n) for n = 1..10011 (T(n,k) for n+k <= 142).
Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
J. C. Kluyver, Some formulae concerning the integers less than n and prime to n, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414; see p. 410.
C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Akad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It may not be universally accessible.]
R. D. von Sterneck, Über ein Analogon zur additiven Zahlentheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113. [Summary of the 1902 paper.]
Wikipedia, Ramanujan's sum.
Wikipedia, Robert Daublebsky von Sterneck der Jüngere.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Transpose of array in A054534. Cf. A054532, A054533, A282634.

Cf. A086831=c_n(2) (2nd column), A085097=c_n(3) (3rd column), A085384=c_n(4) (4th column), A085639=c_n(5) (fifth column), A085906=c_n(6) (sixth column), A099837=c_3(n) (third row), A176742=c_4(n) (fourth row), A100051=c_6(n) (sixth row).

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
# to get the example above
for n to 8 do
    seq(c(n, k), k = 1 .. 13);
end do
# Petros Hadjicostas, Jul 27 2019

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-François Alcover, Nov 10 2011, after Emeric Deutsch *)
(* To get the example above in table format *)
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 13}]]
(* Petros Hadjicostas, Jul 27 2019 *)

T(n,k) = c_n(k) = phi(n) * Moebius(n/gcd(n, k))/phi(n/gcd(n, k)). - Emeric Deutsch, Dec 23 2004 [The r.h.s. of this formula is known as the von Sterneck function, and it was introduced by him around 1900. - Petros Hadjicostas, Jul 20 2019]
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 [We have sigma_{1-s}(k) = Sum_{d|k} d^{1-s} = Sum_{d|k} (k/d)^{1-s} = sigma_{s-1}(k) / k^{s-1}. - Petros Hadjicostas, Jul 27 2019]
From Mats Granvik, Oct 10 2016: (Start)
For n >= 1 and k >= 1 let
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)
T(n,k) = c_n(k) = Sum_{s | gcd(n,k)} s * Moebius(n/s). - Petros Hadjicostas, Jul 27 2019
Lambert series and a consequence: Sum_{n >= 1} c_n(k) * z^n / (1 - z^n) = Sum_{s|k} s * z^s and -Sum_{n >= 1} (c_n(k) / n) * log(1 - z^n) = Sum_{s|k} z^s for |z| < 1 (using the principal value of the logarithm). - Petros Hadjicostas, Aug 15 2019

Name edited by Petros Hadjicostas, Jul 27 2019

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).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 09 2000

The Ramanujan sum is also known as the von Sterneck arithmetic function. Robert Daublebsky von Sterneck introduced it around 1900. - Petros Hadjicostas, Jul 20 2019

T(n, k) = c_k(n) is the sum of the n-th powers of the k-th primitive roots of unity. - Petros Hadjicostas, Jul 27 2019

			Array T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1, -1, -1,  0, -1,  1, -1,  0,  0,  1, -1, ...
  1,  1, -1, -2, -1, -1, -1,  0,  0, -1, -1, ...
  1, -1,  2,  0, -1, -2, -1,  0, -3,  1, -1, ...
  1,  1, -1,  2, -1, -1, -1, -4,  0, -1, -1, ...
  1, -1, -1,  0,  4,  1, -1,  0,  0, -4, -1, ...
  1,  1,  2, -2, -1,  2, -1,  0, -3, -1, -1, ...
  1, -1, -1,  0, -1,  1,  6,  0,  0,  1, -1, ...
  ...

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.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Acad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa).

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
Austrian Biographical Encyclopedia from 1815 onwards, Daublebsky von Sterneck, Robert.
Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
M. V. Subbarao, The Brauer-Rademacher identity, Amer. Math. Monthly 72 (1965), 135-138.
Wikipedia, Ramanujan's sum.
Wikipedia, Robert Daublebsky von Sterneck der Jüngere.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Cf. A000010, A033999, A054532, A054533, A054535, A062570, A085097, A058384, A085639, A085906, A099837, A100051, A176742, A282634.

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 *)
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 11}]] (* to print a table like the one in the example - Petros Hadjicostas, Jul 27 2019 *)

{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 */

{T(n, k) = if( n<1 || k<1, 0, polsym( polcyclo( k), n) [n + 1])} /* Michael Somos, Mar 21 2011 */

/*To get an array like in the example above using Michael Somos' programs:*/
{for (n=1, 20, for (k=1, 40, print1(T(n,k), ","); ); print(); ); } /* Petros Hadjicostas, Jul 27 2019 */

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
Lambert series and a consequence: Sum_{k >= 1} c_k(n) * z^k / (1 - z^k) = Sum_{s|n} s * z^s and -Sum_{k >= 1} (c_k(n) / k) * log(1 - z^k) = Sum_{s|n} z^s for |z| < 1 (using the principal value of the logarithm). - Petros Hadjicostas, Aug 15 2019

A054532 Ramanujan sum T(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2Piimn / k), triangular array read by rows for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 09 2000

T(n, k) = c_k(n) = sum of the n-th powers of the k-th primitive roots of unity. - Petros Hadjicostas, Jul 27 2019

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  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;
  ...

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

T. D. Noe, Rows n=1..50 of triangle, flattened
Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
H. G. Gadiyar and R. Padma, Linking the circle and the sieve: Ramanujan-Fourier series, arXiv:math/0601574 [math.NT], 2006.
Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
J. C. Kluyver, Some formulae concerning the integers less than n and prime to n, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414; see p. 410.
P. Moree and H. Hommerson, Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients, arXiv:math/0307352 [math.NT], 2003.
K. Motose, Ramanujan's sums and cyclotomic polynomials, Math. J. Okayama U. 47, no 1, (2005), Article 5.
C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Akad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It may not be universally accessible.]
R. D. von Sterneck, Über ein Analogon zur additiven Zahlentheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113. [Summary of the 1902 paper.]
Wikipedia, Ramanujan's sum.
Wikipedia, Robert Daublebsky von Sterneck der Jüngere.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Cf. A054533, A054534, A054535, A282634.

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 *)
(* to get the triangle in the example *)
TableForm[Table[t[n, k], {n, 1, 9}, {k, 1, n}]]
(* Petros Hadjicostas, Jul 27 2019 *)

T(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} cos(2*Pi*m*n / k) = mu(k/gcd(k,n)) * phi(k) / phi(k/gcd(k,n)) = Sum_{d | gcd(k,n)} mu(k/d) * d. (All formulas were proved by Kluyver (1906, p. 410).) - Petros Hadjicostas, Aug 20 2019

cp's OEIS Frontend

A054533 Triangular array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) for n >= 1 and 1 <= k <= n.

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A054535 Square array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n), read by antidiagonals upwards (n >= 1, k >= 1).

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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).

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A054532 Ramanujan sum T(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2*Pi*i*m*n / k), triangular array read by rows for n >= 1 and 1 <= k <= n.

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A298983 Triangle read by rows T(n,k) giving coefficients in expansion of Product_{j=1..n} (1-x^j)^2 mod x^(n+1)-1.

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A054532 Ramanujan sum T(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2Piimn / k), triangular array read by rows for n >= 1 and 1 <= k <= n.