A054535 -id:A054535 - cp's OEIS frontend

A177121 Square array T(n,k) read by antidiagonals up: T(n,k) = 1 if n=1; otherwise if n divides k then T(n,k) = -n+1; otherwise T(n,k) = 1.

1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, 1, -1, 1, 1, 1, 1, 1, 1, -2, 1, 1, 1, 1, 1, 1, -4, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -5, 1, -3, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -6, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Mats Granvik and Gary W. Adamson, May 03 2010

Replace the first column in A077049 with any k-th column in A177121 (this array) to get a new array. Then the matrix inverse of the new array will have the k-th column of A054535 (Ramanujan sum) as its first column.

Obtained from A176079 by transposing, flipping signs, and adding a lower triangle of all -1's. - R. J. Mathar, Jul 08 2011

			Table begins:
  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,  1,  1, -3,  1,  1,  1, -3,  1,  1, ...
  1,  1,  1,  1, -4,  1,  1,  1,  1, -4, ...
  1,  1,  1,  1,  1, -5,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1, -6,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1, -7,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, -8,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, -9, ...
  ...

Antti Karttunen, Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)

Cf. A051731, A054535, A077049, A176079.

=if(row()=1;1;if(mod(column();row())=0;-row()+1;1))

up_to = 65703; \\ = binomial(362+1,2)
A177121sq(row,col) = if(1==row,1,if(!(col%row),(1-row),1));
A177121list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A177121sq((a-(col-1)),col))); (v); };
v177121 = A177121list(up_to);
A177121(n) = v177121[n]; \\ Antti Karttunen, Sep 25 2018

T(n,k) = 1 if n=1; otherwise, if n divides k then T(n,k) = -n+1; otherwise T(n,k) = 1.

A127332 A126988 * A002321.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 4, 4, 5, 9, 1, 10, 8, 3, 7, 15, 3, 16, 2, 6, 17, 21, -6, 13, 19, 11, 8, 27, -5, 27, 10, 13, 28, 10, -10, 35, 31, 17, -6, 40, -3, 40, 20, -4, 40, 44, -18, 32, 18, 26, 23, 50, 4, 21, 0, 28, 54, 58, -45, 59, 53, 3, 19, 24, 11, 65, 37, 39, 1

Offset: 1

Gary W. Adamson, Jan 10 2007

sign

			a(6) = 3 = 6*1 + 3*0 + 2*(-1) + 0*(-1) + 0*(-2) + 1*(-1), where (6, 3, 2, 0, 0, 1) = row 6 of A126988.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002321, A054532, A054533, A054534, A054535, A126988.

Block[{nn = 70, m}, m = Table[Sum[MoebiusMu@k, {k, n}], {n, nn}]; Table[Total@ Array[m[[#]] If[Mod[n, #] == 0, n/#, 0] &, n], {n, nn}]] (* Michael De Vlieger, Jun 14 2018 *)

lista(nn) = {mat = matrix(nn, nn, n, k, if (n % k, 0, n/k)); vec = matrix(nn, 1, n, k, if (k==1, mertens(n), 0)); res = (mat*vec); for (n = 1, nn, print1(res[n, 1], ", "););} \\ Michel Marcus, Sep 25 2013

a(n) = sum(k=1, n, moebius(k / gcd(n, k)) * eulerphi(k) / eulerphi(k / gcd(n, k))); \\ Daniel Suteu, Jun 23 2018

M * V where M = A126988 as an infinite lower triangular matrix and V = the Mertens sequence, A002321 as a vector: [1, 0, -1, -1, -2, -1, ...].

a(n) = Sum_{q=1..n} c_q(n), where c_q(n) is the Ramanujan's sum function given in A054533. - Daniel Suteu, Jun 14 2018

Corrected and extended by Michel Marcus, Sep 25 2013

A159936 Triangle read by rows, A051731 * A054533 * transpose(A101688), provided A101688 is read as a square array.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 2, 3, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 2, 4, 4, 4, 4, 1, 1, 2, 3, 3, 3, 6, 6, 6, 1, 1, 2, 2, 4, 4, 5, 4, 5, 4, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 2, 2, 3, 2, 4, 4, 6, 6, 4, 4, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 1, 2, 2, 4, 4, 6, 6, 7, 6, 7, 6, 7, 6

Offset: 1

Gary W. Adamson, Apr 26 2009

Row sums = A057661: (1, 2, 4, 6, 11, 11, 22,...). Right border = A000010, phi(n).

			First few rows of the triangle are as follows:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 2;
  1, 1, 2, 3, 4;
  1, 1, 2, 2, 3, 2;
  1, 1, 2, 3, 4, 5, 6;
  1, 1, 2, 2, 4, 4, 4, 4;
  1, 1, 2, 3, 3, 3, 6, 6, 6;
  1, 1, 2, 2, 4, 4, 5, 4, 5, 4;
  1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
  1, 1, 2, 2, 3, 2, 4, 4, 6, 6,  4,  4;
  1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
  1, 1, 2, 2, 4, 4, 6, 6, 7, 6,  7,  6,  7, 6;
  1, 1, 2, 3, 3, 3, 5, 4, 3, 5,  9,  8, 10, 9, 8;
  1, 1, 2, 2, 4, 4, 4, 4, 8, 8,  8,  8,  8, 8, 8, 8;
  ...

Cf. A000010, A051731, A054532, A054533, A054534, A054535, A057661, A101688.

Triangle read by rows, A051731 * A054533 * A000012. A051731 = the inverse Mobius transform. A054533 = the lower left half of the Ramanujan sum table. The operation (* transpose(A101688)) takes partial sums of (A051731 * A054533) starting from the right. [Edited by Petros Hadjicostas, Jul 30 2019]

Name edited by Petros Hadjicostas, Jul 30 2019

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.

Original entry on oeis.org

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

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A177121 Square array T(n,k) read by antidiagonals up: T(n,k) = 1 if n=1; otherwise if n divides k then T(n,k) = -n+1; otherwise T(n,k) = 1.

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A127332 A126988 * A002321.

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A159936 Triangle read by rows, A051731 * A054533 * transpose(A101688), provided A101688 is read as a square array.

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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.

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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.

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

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