A085906 -id:A085906 - cp's OEIS frontend

A067856 Sum_{n >= 1} a(n)/n^s = 1/(Sum_{n >= 1} (-1)^(n + 1)/n^s).

1, 1, -1, 2, -1, -1, -1, 4, 0, -1, -1, -2, -1, -1, 1, 8, -1, 0, -1, -2, 1, -1, -1, -4, 0, -1, 0, -2, -1, 1, -1, 16, 1, -1, 1, 0, -1, -1, 1, -4, -1, 1, -1, -2, 0, -1, -1, -8, 0, 0, 1, -2, -1, 0, 1, -4, 1, -1, -1, 2, -1, -1, 0, 32, 1, 1, -1, -2, 1, 1, -1, 0, -1, -1, 0, -2, 1, 1, -1, -8, 0, -1, -1, 2, 1, -1, 1, -4, -1, 0, 1, -2, 1, -1, 1, -16, -1, 0, 0, 0

Offset: 1

Leroy Quet, Feb 15 2002

Dirichlet inverse of A062157. - R. J. Mathar, Jul 15 2010
The first 31 terms equal the values of the Ramanujan sum c_n(8) -- see for example A085906 -- but a(32) <> c_{32}(8). - R. J. Mathar, Apr 02 2011
From  Peter Bala, Mar 12 2019: (Start)
Let Mu(n) = (-1)^(n+1)*a(n), an analog of the Möbius function mu(n). Then for arithmetic functions f(n) and g(n) we have the following analog of the Möbius inversion formula: f(n) = Sum_{d divides n} (-1)^((1+d)*(1+n/d))*g(d) iff g(n) = Sum_{d divides n} (-1)^((1+d)*(1+n/d))*Mu(n/d)*f(d).
Each of the following two equations implies the other: F(x) = Sum_{n >= 1} (-1)^(n+1)*G(n*x); G(x) = Sum_{n >= 1} a(n)*F(n*x). See G. Pólya and G. Szegő, Part V111, Chap. 1, Nos. 66-68.2. (End)
Let D(n) denote the set of partitions of n into distinct parts. Then Sum_{parts k in D(n)} a(k) = |D(n-1)| = A000009(n-1). For example, D(6) = {6, 1 + 5, 2 + 4, 1 + 2 + 3} and the sum a(6) + (a(1) + a(5)) + (a(2) + a(4)) + (a(1) + a(2) + a(3)) = 3 = |D(5)|. - Peter Bala, Mar 14 2019
From Petros Hadjicostas, Jul 25 2020: (Start)
For p prime >= 2, Petrogradsky (2003) defined the multiplicative functions 1_p and mu_p in the following way:
1_p(n) = 1 when gcd(n,p) = 1 and 1_p(n) = 1 - p when gcd(n,p) = p;
mu_p(n) = mu(n) when gcd(n,p) = 1 and mu_p(n) = mu(m)*(p^s - p^(s-1)) when n = m*p^s with gcd(m,p) = 1 and s >= 1.
We have 1_2(n) = A062157(n), 1_3(n) = A061347(n) (with offset 1), a(n) = mu_2(n), and A117997(n) = mu_3(n) for n >= 1.
Some of the results by other contributors can be generalized:
(i) Rogel's (1897) formula becomes Sum_{d | n} 1_p(d) * mu_p(n/d) = 0 for n > 1. Thus, 1_p is the Dirichlet inverse of mu_p.
(ii) R. J. Mathar's Dirichlet g.f. for mu_p becomes 1/(zeta(s) * (1 - p^(1-s))). The Dirichlet g.f. for 1_p is zeta(s) * (1 - p^(1-s)).
(iii) Benoit Cloitre's formula becomes 1 = Sum_{k=1..n} mu_p(k)*g_p(n/k), where g_p(x) = floor(x) - p*floor(x/p) = floor(x) mod p.
(iv) Paul D. Hanna's formula becomes Sum_{n >= 1} (mu_p(n)/n)*log((1 - x^(n*p))/(1 - x^n)) = x.
(v) The definition of A117997 generalizes to Sum_{d | n} mu_p(d) = n, if n = p^s for s >= 0, and = 0, otherwise.
(vi) By differentiating both sides of (iv) w.r.t. x and multiplying both sides by x, we get Sum_{n >= 1} mu_p(n)*(x^n + 2*x^(2*n) + ... + (p-1)*x^(n*(p-1)))/(1 + x^n + x^(2*n) + ... + x^(n*(p-1))) = x, which generalizes one of Peter Bala's formulas. It can be thought as a "generalized Lambert series".
(vii) Obviously, f(n) = Sum_{d | n} 1_p(n/d)*g(d) if and only if g(n) = Sum_{d | n} mu_p(n/d)*f(d). (End)

G. Pólya and G. Szegő, Problems and Theorems in Analysis Volume II. Springer_Verlag 1976.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Peter Bala, A signed Dirichlet product of arithmetical functions
V. M. Petrogradsky, Witt's formula for restricted Lie algebras, Advances in Applied Mathematics, 30 (2003), 219-227.
V. M. Petrogradsky, Witt's formula for restricted Lie algebras, Advances in Applied Mathematics, 30 (2003), 219-227.
Franz Rogel, Transformationen arithmetischer Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article LI (1897), Prague (31 pages); see pp. 10-11 and especially Eqs. (21) - (24). [There are some obvious typos there; especially Eq. (24) should become Sum_{t | v} (-1)^(v/t) * c(t) = 0 for v > 1, which is the equation a(n) = Sum{k | n, 1 < k} (-1)^k a(n/k), for n >= 2, in the Formula section below. - _Petros Hadjicostas_, Jul 21 2019]

Cf. A000009, A038712, A038838, A048298 (inverse Mobius transform), A061347, A062157, A085906, A117997, A321088 (Euler transform), A321558.

a[n_] := If[n == 1, 1, Product[{p, e} = pe; If[2 == p, e--, If[e > 1, p = 0, p = -1]]; p^e, {pe, FactorInteger[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Oct 01 2019 *)

{a(n)=local(k); if(n<1, 0, k=valuation(n, 2); moebius(n/2^k)*2^max(0, k-1))} /* Michael Somos, Aug 22 2006 */

A067856(n) =  { my(f=factor(n)); for(i=1,#f~,if(2==f[i,1],f[i,2]--,if(f[i,2]>1,f[i,1]=0,f[i,1]=-1))); factorback(f); }; \\ Antti Karttunen, Sep 27 2018, after Vladeta Jovovic_'s multiplicative formula.

a(1) = 1 and a(n) = Sum{k | n, 1 < k} (-1)^k a(n/k) for n >= 2; the sum is over the divisors, k, of n, where k > 1. If n is odd, a(n) = mu(n), where mu(.) is the Moebius function. If n is even, a(n) = mu(m)* 2^(k-1), where n = m*2^k, m is odd integer, and k is a positive integer.
Sum_{n > 0} a(n)*x^n/(1 + x^n) = x. Moebius transform of A048298. Multiplicative with a(2^e) = 2^(e - 1), a(p) = -1 for p > 2, a(p^e) = 0 for p > 2 and e > 1. - Vladeta Jovovic, Jan 02 2003
Sum_{n > 0} a(n)*log(1 + x^n)/n = x. - Paul D. Hanna, May 06 2003
a(n) = 0 if and only if n is divisible by the square of an odd prime (A038838). - Michael Somos, Aug 22 2006
1 = Sum_{k=1..n} a(k)*g(n/k), where g(x) = floor(x) - 2*floor(x/2). - Benoit Cloitre, Nov 11 2010
Dirichlet g.f.: 1/( zeta(s) * (1 - 2^(1-s)) ). - R. J. Mathar, Apr 02 2011
From Peter Bala, Mar 13 2019: (Start)
Sum_{n >= 1} a(n)*x^n/(1 + x^n) = x
Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x + 2*x^2 + 4*x^4 + 8*x^8 + 16*x^16 + ...
Sum_{n >= 1} a(n)*x^n/(1 + (-x)^n) = x + 2*(x^2 + x^4 + x^8 + x^16 + ...)
Sum_{n >= 1} a(n)*x^n/(1 - (-x)^n) = x + 2*(x^4 + 3*x^8 + 7*x^16 + 15*x^32 + ...). (End)
G.f. A(x) satisfies: A(x) = x + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... - Ilya Gutkovskiy, May 11 2019
Sum_{k=1..n} abs(a(k)) ~ 2*n*(log(n) - 1 + gamma + 11*log(2)/6 - 12*zeta'(2)/Pi^2) / (log(2)*Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 30 2024

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

A085097 Ramanujan sum c_n(3).

Original entry on oeis.org

1, -1, 2, 0, -1, -2, -1, 0, -3, 1, -1, 0, -1, 1, -2, 0, -1, 3, -1, 0, -2, 1, -1, 0, 0, 1, 0, 0, -1, 2, -1, 0, -2, 1, 1, 0, -1, 1, -2, 0, -1, 2, -1, 0, 3, 1, -1, 0, 0, 0, -2, 0, -1, 0, 1, 0, -2, 1, -1, 0, -1, 1, 3, 0, 1, 2, -1, 0, -2, -1, -1, 0, -1, 1, 0, 0, 1, 2, -1, 0, 0, 1, -1, 0, 1, 1, -2, 0, -1, -3, 1, 0, -2, 1, 1, 0, -1, 0, 3, 0, -1, 2, -1, 0, 2, 1, -1, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 10 2003

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Acad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It seems that his father, Robert Freiherr Daublebsky von Sterneck, had exactly the same name.]

Antti Karttunen, Table of n, a(n) for n = 1..65537
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.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
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.
Wikipedia, Ramanujan's sum.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Cf. A000010, A008683, A086831, A085906.

f[list_, i_] := list[[i]]; nn = 105; a =Table[MoebiusMu[n], {n, 1, nn}]; b =Table[If[IntegerQ[3/n], n, 0], {n, 1, nn}];Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Dec 30 2015 *)
f[3, e_] := Switch[e, 1, 2, 2, -3, , 0]; f[p, e_] := If[e == 1, -1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2023 *)

a(n)=eulerphi(n)*moebius(n/gcd(n,3))/eulerphi(n/gcd(n,3))

a(n) = phi(n)*mu(n/gcd(n, 3)) / phi(n/gcd(n, 3)).
Dirichlet g.f.: (1+3^(1-s))/zeta(s). [Titchmarsh eq. (1.5.4.)] - R. J. Mathar, Mar 26 2011
Multiplicative with a(3) = 2, a(3^2) = -3, a(3^e) = 0 for e >= 3, for a prime p != 3, a(p) = -1 and a(p^e) = 0 for e >= 2. - Amiram Eldar, Sep 10 2023
Sum_{k=1..n} abs(a(k)) ~ (9/Pi^2) * n. - Amiram Eldar, Jan 21 2024

More terms from Benoit Cloitre, Aug 12 2003

A086831 Ramanujan sum c_n(2).

Original entry on oeis.org

1, 1, -1, -2, -1, -1, -1, 0, 0, -1, -1, 2, -1, -1, 1, 0, -1, 0, -1, 2, 1, -1, -1, 0, 0, -1, 0, 2, -1, 1, -1, 0, 1, -1, 1, 0, -1, -1, 1, 0, -1, 1, -1, 2, 0, -1, -1, 0, 0, 0, 1, 2, -1, 0, 1, 0, 1, -1, -1, -2, -1, -1, 0, 0, 1, 1, -1, 2, 1, 1, -1, 0, -1, -1, 0, 2, 1, 1, -1, 0, 0, -1, -1, -2, 1, -1, 1, 0, -1, 0, 1, 2, 1, -1, 1, 0, -1, 0, 0, 0, -1, 1, -1, 0, -1

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 07 2003

Mobius transform of 1,2,0,0,0,0,... (A130779). - R. J. Mathar, Mar 24 2012

			a(4) = -2 because the primitive fourth roots of unity are i and -i.  We sum their squares to get i^2 + (-i)^2 = -1 + -1 = -2. - _Geoffrey Critzer_, Dec 30 2015

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd edn., 1986.

Antti Karttunen, Table of n, a(n) for n = 1..65537
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.
Michael L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
Charles 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.
Charles 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.
Wikipedia, Ramanujan's sum.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Cf. A000010, A008683, A054532, A054533, A054534, A054535.

Cf. A085097, A085384, A085639, A085906 for Ramanujan sums c_n(3), c_n(4), c_n(5), c_n(6).

with(numtheory):a:=n->phi(n)*mobius(n/gcd(n,2))/phi(n/gcd(n,2)): seq(a(n),n=1..130); # Emeric Deutsch, Dec 23 2004

f[list_, i_] := list[[i]]; nn = 105; a = Table[MoebiusMu[n], {n, 1, nn}]; b =Table[If[IntegerQ[2/n], n, 0], {n, 1,nn}];Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Dec 30 2015 *)
f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := Switch[e, 1, 1, 2, -2, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 21 2024 *)

A086831(n) = (eulerphi(n)*moebius(n/gcd(n, 2))/eulerphi(n/gcd(n, 2))); \\ Antti Karttunen, Sep 27 2018

For a general k >= 1, c_n(k) = phi(n)*mu(n/gcd(n, k)) / phi(n/gcd(n, k)); so c_n(1) = mu(n) = A008683(n).
a(n) = phi(n)*mu(n/gcd(n, 2)) / phi(n/gcd(n, 2)).
Dirichlet g.f.: (1+2^(1-s))/zeta(s). [Titchmarsh eq. (1.5.4)] - R. J. Mathar, Mar 26 2011
Multiplicative with a(2) = 1, a(2^2) = -2, and a(2^e) = 0 for e >= 3, and for an odd prime p, a(p) = -1 and a(p^e) = 0 for e >= 2. - Amiram Eldar, Sep 14 2023
Sum_{k=1..n} abs(a(k)) ~ (8/Pi^2) * n. - Amiram Eldar, Jan 21 2024

Corrected and extended by Emeric Deutsch, Dec 23 2004

A085639 Ramanujan sum c_n(5).

Original entry on oeis.org

1, -1, -1, 0, 4, 1, -1, 0, 0, -4, -1, 0, -1, 1, -4, 0, -1, 0, -1, 0, 1, 1, -1, 0, -5, 1, 0, 0, -1, 4, -1, 0, 1, 1, -4, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 5, 1, 0, -1, 0, -4, 0, 1, 1, -1, 0, -1, 1, 0, 0, -4, -1, -1, 0, 1, 4, -1, 0, -1, 1, 5, 0, 1, -1, -1, 0, 0, 1, -1, 0, -4, 1, 1, 0, -1

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 15 2003

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.

Antti Karttunen, Table of n, a(n) for n = 1..65537
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.
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.
M. V. Subbarao, The Brauer-Rademacher identity, Amer. Math. Monthly 72 (1965), 135-138.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
Wikipedia, Ramanujan's sum.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Cf. A000010, A008683, A054532, A054533, A054534, A054535, A085384, A085906, A085097, A086831.

a[n_] := EulerPhi[n] * MoebiusMu[n/GCD[n, 5]] / EulerPhi[n/GCD[n, 5]]; Table[ a[n], {n, 1, 105}]
f[p_, e_] := If[e == 1, -1, 0]; f[5, e_] := Switch[e, 1, 4, 2, -5, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 21 2024 *)

a(n)=eulerphi(n)*moebius(n/gcd(n,5))/eulerphi(n/gcd(n,5))

a(n) = phi(n)*mu(n/gcd(n, 5)) / phi(n/gcd(n, 5)).
Dirichlet g.f.: (1+5^(1-s))/zeta(s). - R. J. Mathar, Mar 26 2011
Lambert series and a consequence: Sum_{n >= 1} c_n(5) * z^n / (1 - z^n) = z + 5*z^5 and -Sum_{n >= 1} (c_n(5) / n) * log(1 - z^n) = z + z^5 for |z| < 1 (using the principal value of the logarithm). - Petros Hadjicostas, Aug 24 2019
From Amiram Eldar, Jan 21 2024: (Start)
Multiplicative with a(5) = 4, a(5^2) = -5, and a(5^e) = 0 for e >= 3, and for a prime p != 5, a(p) = -1, and a(p^e) = 0 for e >= 2.
Sum_{k=1..n} abs(a(k)) ~ (10/Pi^2) * n. (End)

More terms from Robert G. Wilson v and Benoit Cloitre, Aug 17 2003

cp's OEIS Frontend

A067856 Sum_{n >= 1} a(n)/n^s = 1/(Sum_{n >= 1} (-1)^(n + 1)/n^s).

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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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A085097 Ramanujan sum c_n(3).

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A086831 Ramanujan sum c_n(2).

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A085639 Ramanujan sum c_n(5).

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