A054532 -id:A054532 - cp's OEIS frontend

A227507 Table of p(a,n) read by antidiagonals, where p(a,n) = Sum_{k=1..n} gcd(k,n) exp(2 Pi i k a / n) is the Fourier transform of the greatest common divisor.

1, 3, 1, 5, 1, 1, 8, 2, 3, 1, 9, 2, 2, 1, 1, 15, 4, 4, 5, 3, 1, 13, 2, 4, 2, 2, 1, 1, 20, 6, 6, 4, 8, 2, 3, 1, 21, 4, 6, 5, 4, 2, 5, 1, 1, 27, 6, 8, 6, 6, 9, 4, 2, 3, 1, 21, 4, 6, 4, 6, 2, 4, 2, 2, 1, 1, 40, 10, 12, 12, 12, 6, 15, 4, 8, 5, 3, 1, 25, 4, 10, 4, 6, 4, 6, 2, 4, 2, 2, 1, 1, 39, 12, 8, 10, 12, 6

Offset: 1

Peter H van der Kamp, Jul 13 2013

p(a,n) gives the number of pairs (i,j) of congruence classes modulo n, such that i*j = a mod n.

p(a,n) is a multiplicative function of n.

			1, 3, 5, 8, 9, 15, 13, 20, 21, 27
1, 1, 2, 2, 4, 2, 6, 4, 6, 4
1, 3, 2, 4, 4, 6, 6, 8, 6, 12
1, 1, 5, 2, 4, 5, 6, 4, 12, 4
1, 3, 2, 8, 4, 6, 6, 12, 6, 12
1, 1, 2, 2, 9, 2, 6, 4, 6, 9
The array G_d(n) of Abel et al. (with A018804 on the diagonal) starts as follows:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ,...
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3,...
2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2,...
2, 4, 2, 8, 2, 4, 2, 8, 2, 4, 2, 8, 2, 4, 2, 8, 2, 4, 2, 8,...
4, 4, 4, 4, 9, 4, 4, 4, 4, 9, 4, 4, 4, 4, 9, 4, 4, 4, 4, 9,...
2, 6, 5, 6, 2,15, 2, 6, 5, 6, 2,15, 2, 6, 5, 6, 2,15, 2, 6,...
6, 6, 6, 6, 6, 6,13, 6, 6, 6, 6, 6, 6,13, 6, 6, 6, 6, 6, 6,...
4, 8, 4,12, 4, 8, 4,20, 4, 8, 4,12, 4, 8, 4,20, 4, 8, 4,12,..
6, 6,12, 6, 6,12, 6, 6,21, 6, 6,12, 6, 6,12, 6, 6,21, 6, 6,...
4,12, 4,12, 9,12, 4,12, 4,27, 4,12, 4,12, 9,12, 4,12, 4,27,...
10,10,10,10,10,10,10,10,10,10,21,10,10,10,10,10,10,10,10,10,...
4, 8,10,16, 4,20, 4,16,10, 8, 4,40, 4, 8,10,16, 4,20, 4,16,...
12,12,12,12,12,12,12,12,12,12,12,12,25,12,12,12,12,12,12,12,...
... - _R. J. Mathar_, Jan 21 2018

U. Abel, W. Awan, and V. Kushnirevych, A Generalization of the Gcd-Sum Function, J. Int. Seq. 16 (2013), #13.6.7.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, INTEGERS 13 (2013), A24.

Cf. A000010, A018804, A054532, A054533, A054534, A054535.

p:=(a,n)->add(d*phi(n/d),d in divisors(gcd(a,n))):
seq(seq(p(a,n-a),a=0..n-1),n=1..10);

The function can be written as a generalized Ramanujan sum: p(a,n) = Sum_{d|gcd(a,n)} d phi(n/d), where phi(n) denotes the totient function.
The rows of its table are equal to two of the diagonals: p(a,n) = p(n-a,n) = p(n+a,n).
p(0,n) = A018804(n), p(1,n) = A000010(n).
f(n) = Sum_{k=1..n} p(r,k)/k = Sum_{k=1..n} c_k(r)/k * floor(n/k), where c_k(r) denotes Ramanujan's sum (A054533(r)).

A086811 Average (scaled by a certain explicit factor) over all integers k of a_k(n), the n-th coefficient of the k-th cyclotomic polynomial.

Original entry on oeis.org

0, 3, 6, 16, 45, 126, 224, 1344, 684, 1116, 4752, 23760, 56784, 286944, 164664, 281472, 2449224, 7371648, 27086400, 160392960, 49635936, 68277888, 1049956992, 6077306880, 1252224000, 3240801792, 2083408128, 4066530048, 35225729280, 142745587200, 717382656000, 6279166033920, 2442775449600, 2080906813440, 2251759104000

Offset: 1

Pieter Moree (moree(AT)mpim-bonn.mpg.de), Aug 05 2003

When n is odd the n-th term is an integer. If n is even then twice the n-th term is an integer. Conjecturally (Y. Gallot) the n-th term is always an integer. For n <= 128 this has been verified numerically by Yves Gallot. It is also an unproved conjecture due to H. Möller (1970) that no term of this sequence is negative.

Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
Tom M. Apostol, The resultant of the cyclotomic polynomials F_m(ax) and F_n(bx), Math. Comp. 29 (1975), 1-6.
Gennady Bachman, On the coefficients of cyclotomic polynomials, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1991, 86 pp.
Gennady Bachman, On the coefficients of cyclotomic polynomials, Mem. Amer. Math. Soc. 106 (1993), no. 510, 80 pp.
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.
Yves Gallot, Pieter Moree, and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients, arXiv:0803.2483 [math.NT], 2008.
Yves Gallot, Pieter Moree, and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients, Unif. Distrib. Theory 6 (2011), 177-206.
Sherry Gong, On a problem regarding coefficients of cyclotomic polynomials, J. Number Theory 129 (2009), 2924-2932.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
G. S. Kazandzidis, On the cyclotomic polynomial: Coefficients, Bull. Soc. Math. Grèce (N.S.) 4A (1963), 1-11.
G. S. Kazandzidis, On the cyclotomic polynomials: Morphology-Estimates, Bull. Soc. Math. Grèce (N.S.) 4A (1963), 50-73.
D. H. Lehmer, Some properties of the cyclotomic polynomial, J. Math. Anal. Appl. 15 (1966), 105-117.
H. Möller, Über die i-ten Koeffizienten der Kreisteilungspolynome, Math. Ann. 188 (1970), 26-38.
Pieter Moree and Huib Hommersom, 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.
Wikipedia, Cyclotomic polynomial.
Wikipedia, Ramanujan's sum.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Cf. A013595, A013596, A054532, A054533, A054534, A054535.

with(numtheory):for k from 1 to 50 do; v := 1: w := 1:j := 1:z := 1:while ithprime(j)<=k do; v := v*ithprime(j); w := w*(1+1/ithprime(j)); z := z*(ithprime(j)+1); j := j+1; end do: v := v*k:z := z*k:q := ithprime(j):te := 0:for i from 1 to nops(divisors(v)) do; d := divisors(v)[i]; kl(x) := 1; for j from 1 to k do; if modp(d,j)=0 then kl(x) := taylor(kl(x)*(1-x^j)^mobius(d/j),x,k+1); end if; end do: te := te+coeff(kl(x),x,k)/d; kl(x) := 1; for j from 1 to k do; if modp(q*d,j)=0 then kl(x) := taylor(kl(x)*(1-x^j)^mobius(q*d/j),x,k+1); end if; end do: te := te+coeff(kl(x),x,k)/d; end do: zr := te/(2*w):print(k,zr*z):end do:

Let M_k = k * Product_{prime p<=k} p. Let q be any prime > k. Then the k-th term (for k >= 2) is M_k * Sum_{d|M_k} ( a_d(k) + a_{d*q}(k) )/(2*d). The average of the k-th coefficient of the n-th cyclotomic polynomial is given by the k-th coefficient of this sequence divided by Zeta(2) * k * Product_{p<=k} (p+1). (Zeta(2) = Pi^2/6.) [See Section 8.3 in Moree and Hommerson (2003).]

More terms from Petros Hadjicostas, Aug 01 2019 using the author's Maple program

A144733 Triangle read by rows, 2*A054533 - A054521.

Original entry on oeis.org

1, -3, 2, -3, -3, 4, -1, -4, -1, 4, -3, -3, -3, -3, 8, 1, -2, -4, -2, 1, 4, -3, -3, -3, -3, -3, -3, 12, -1, 0, -1, -8, -1, 0, -1, 8, -1, -1, -6, -1, -1, -6, -1, -1, 12, 1, -2, 1, -2, -8, -2, 1, -2, 1, 8, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, 20

Offset: 1

Gary W. Adamson, Sep 20 2008

Right border = A140434: (1, 2, 4, 4, 8, 4, 12,...).
Left border = A133695: (1, -3, -3, -1, -3, 1, -3, -1,...)
Row sums = A000010, with negative signs after the first 1: (1, -1, -2, -2, -4, -2, -6,...).

			First few rows of the triangle =
   1;
  -3,  2;
  -3, -3,  4;
  -1, -4, -1,  4;
  -3, -3, -3, -3,  8;
   1, -2, -4, -2,  1,  4;
  -3, -3, -3, -3, -3, -3, 12;
  -1,  0, -1, -8  -1,  0, -1,  8;
  -1, -1, -6, -1, -1, -6, -1, -1, 12;
   1, -2,  1, -2, -8, -2,  1, -2,  1,  8;
  -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, 20;
   ...

Cf. A000010, A054521, A054532, A054533, A054534, A054535, A140434, A133695.

Triangle read by rows, 2*A054533 - A054521; as infinite lower triangular matrices.

T(n,k) = -I(gcd(n,k) = 1) + 2 * Sum_{d|gcd(n,k)} d * mu(n/d) for n >= 1 and 1 <= k <= n, where I(condition) = 1 if the condition holds, and 0 otherwise. - Petros Hadjicostas, Jul 29 2019

cp's OEIS Frontend

A227507 Table of p(a,n) read by antidiagonals, where p(a,n) = Sum_{k=1..n} gcd(k,n) exp(2 Pi i k a / n) is the Fourier transform of the greatest common divisor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A086811 Average (scaled by a certain explicit factor) over all integers k of a_k(n), the n-th coefficient of the k-th cyclotomic polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

Extensions

A144733 Triangle read by rows, 2*A054533 - A054521.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula