A164306 -id:A164306 - cp's OEIS frontend

A054521 Triangle read by rows: T(n,k) = 1 if gcd(n, k) = 1, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane, Apr 09 2000

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, ...). - Gary W. Adamson, May 20 2007
Characteristic function of A169581: a(A169581(n)) = 1; a(A169582(n)) = 0. - Reinhard Zumkeller, Dec 02 2009
The function T(n,k) = T(k,n) is defined for k > n but only the values for 1 <= k <= n as a triangular array are listed here.
T(n,k) = |K(n-k|k)| where K(i|j) is the Kronecker symbol. - Peter Luschny, Aug 05 2012
Twice the sum over the antidiagonals, starting with entry T(n-1,1), for n >= 3, is the same as the row n sum (i.e., phi(n): 2*Sum_{k=1..floor(n/2)} T(n-k,k) = phi(n), n >= 3). - Wolfdieter Lang, Apr 26 2013
The number of zeros in the n-th row of the triangle is cototient(n) = A051953(n). - Omar E. Pol, Apr 21 2017
This triangle is the j = 1 sub-triangle of A349221(n,k) = Sum_{j>=1} [k|binomial(n-1,k-1) AND gcd(n,k) = j], n >= 1, 1 <= k <= n, where [] is the Iverson bracket. - Richard L. Ollerton, Dec 14 2021

			The triangle T(n,k) begins:
  n\k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
   1:  1
   2:  1  0
   3:  1  1  0
   4:  1  0  1  0
   5:  1  1  1  1  0
   6:  1  0  0  0  1  0
   7:  1  1  1  1  1  1  0
   8:  1  0  1  0  1  0  1  0
   9:  1  1  0  1  1  0  1  1  0
  10:  1  0  1  0  0  0  1  0  1  0
  11:  1  1  1  1  1  1  1  1  1  1  0
  12:  1  0  0  0  1  0  1  0  0  0  1  0
  13:  1  1  1  1  1  1  1  1  1  1  1  1  0
  14:  1  0  1  0  1  0  0  0  1  0  1  0  1  0
  15:  1  1  0  1  0  0  1  1  0  0  1  0  1  1  0
  ... (Reformatted by _Wolfdieter Lang_, Apr 26 2013)
Sums over antidiagonals: n = 3: 2*T(2,1) = 2 = T(3,1) + T(3,2) = phi(3). n = 4: 2*(T(3,1) + T(2,2)) = 2 = phi(4), etc. - _Wolfdieter Lang_, Apr 26 2013

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Jakub Jaroslaw Ciaston, A054531 vs A164306 (plot shows these ones)
Index entries for characteristic functions

Cf. A051731, A054522, A215200.
Cf. A050873, A063524.
Cf. A349221.

a054521 n k = a054521_tabl !! (n-1) !! (k-1)
a054521_row n = a054521_tabl !! (n-1)
a054521_tabl = map (map a063524) a050873_tabl
a054521_list = concat a054521_tabl
-- Reinhard Zumkeller, Sep 03 2015

A054521_row := n -> seq(abs(numtheory[jacobi](n-k,k)),k=1..n);
for n from 1 to 13 do A054521_row(n) od; # Peter Luschny, Aug 05 2012

T[ n_, k_] := Boole[ n>0 && k>0 && GCD[ n, k] == 1] (* Michael Somos, Jul 17 2011 *)
T[ n_, k_] := If[ n<1 || k<1, 0, If[ k>n, T[ k, n], If[ k==1, 1, If[ n>k, T[ k, Mod[ n, k, 1]], 0]]]] (* Michael Somos, Jul 17 2011 *)

{T(n, k) = n>0 && k>0 && gcd(n, k)==1} /* Michael Somos, Jul 17 2011 */

def A054521_row(n): return [abs(kronecker_symbol(n-k,k)) for k in (1..n)]
for n in (1..13): print(A054521_row(n)) # Peter Luschny, Aug 05 2012

T(n,k) = A063524(A050873(n,k)). - Reinhard Zumkeller, Dec 02 2009, corrected Sep 03 2015

T(n,k) = A054431(n,k) = A054431(k,n). - R. J. Mathar, Jul 21 2016

A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

Original entry on oeis.org

1, 2, 4, 6, 11, 11, 22, 22, 31, 32, 56, 39, 79, 65, 74, 86, 137, 92, 172, 116, 151, 167, 254, 151, 261, 236, 274, 237, 407, 221, 466, 342, 389, 410, 452, 336, 667, 515, 550, 452, 821, 452, 904, 611, 641, 761, 1082, 599, 1051, 782, 956, 864, 1379, 821, 1166

Offset: 1

Henry Gould, Oct 15 2000

Sum of numerators of n-th order Farey series (cf. A006842). - Benoit Cloitre, Oct 28 2002
Equals row sums of triangle A143613. - Gary W. Adamson, Aug 27 2008
Equals row sums of triangle A159936. - Gary W. Adamson, Apr 26 2009
Also row sums of triangle A164306. - Reinhard Zumkeller, Aug 12 2009

H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), 39 (1997), 11-35.
H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), 39 (1997), 183-194.

T. D. Noe, Table of n, a(n) for n = 1..1000
Zachary Franco, Problem 12114, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 469; A Dirichlet Series with Reduced Numerators, Solution to Problem 12114 by Tamas Wiandt, ibid., Vol. 128, No. 1 (2021), pp. 91-92.
Index entries for sequences related to lcm's.

Cf. A000010, A000217, A006842, A018804, A023896, A051193, A057660, A143613, A159936, A164306.

See A341316 for another version.

a057661 n = a051193 n `div` n  -- Reinhard Zumkeller, Jun 10 2015

[&+[&+[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(n)]: n in [1..100]]; // Jaroslav Krizek, Dec 28 2016

Table[Total[Numerator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; Array[a, 100] (* Amiram Eldar, Apr 26 2023 *)

a(n)=sum(k=1,n,lcm(n,k))/n \\ Charles R Greathouse IV, Feb 07 2017

from math import lcm
def A057661(n): return sum(lcm(n,k)//n for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A057661(n): return 1+prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items())>>1 # Chai Wah Wu, Aug 05 2024

a(n) = (1+A057660(n))/2.
a(n) = A051193(n)/n.
a(n) = Sum_{d|n} psi(d), where psi(m) = is the sum of totatives of m (A023896). - Jaroslav Krizek, Dec 28 2016
a(n) = Sum_{i=1..n} denominator(n/i). - Wesley Ivan Hurt, Feb 26 2017
G.f.: x/(2*(1 - x)) + (1/2)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, Aug 31 2017
If p is prime, then a(p) = T(p-1) + 1 = p(p-1)/2 + 1, where T(n) = n(n+1)/2 is the n-th triangular number (A000217). - David Terr, Feb 10 2019
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, May 29 2021
Dirichlet g.f.: zeta(s)*(1 + zeta(s-2)/zeta(s-1))/2 (Franco, 2019). - Amiram Eldar, Mar 26 2022

More terms from James Sellers, Oct 16 2000

A054531 Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 2, 4, 1, 5, 5, 5, 5, 1, 6, 3, 2, 3, 6, 1, 7, 7, 7, 7, 7, 7, 1, 8, 4, 8, 2, 8, 4, 8, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 12, 6, 4, 3, 12, 2, 12, 3, 4, 6, 12, 1, 13, 13, 13, 13, 13

Offset: 1

N. J. A. Sloane, Apr 09 2000

Sum of n-th row = A057660(n). - Reinhard Zumkeller, Aug 12 2009
Read as a linear sequence, this is conjectured to be the length of the shortest cycle of pebble-moves among the partitions of n (cf. A201144). - Andrew V. Sutherland, Nov 27 2011
The triangle of fractions A226314(i,j)/A054531(i,j) is an efficient way to enumerate the rationals [Fortnow]. - N. J. A. Sloane, Jun 09 2013

			Triangle begins
   1;
   2,  1;
   3,  3,  1;
   4,  2,  4,  1;
   5,  5,  5,  5,  1;
   6,  3,  2,  3,  6,  1;
   7,  7,  7,  7,  7,  7,  1;
   8,  4,  8,  2,  8,  4,  8,  1;
   9,  9,  3,  9,  9,  3,  9,  9,  1;
  10,  5, 10,  5,  2,  5, 10,  5, 10,  1;
  11, 11, 11, 11, 11, 11, 11, 11, 11, 11,  1;
  12,  6,  4,  3, 12,  2, 12,  3,  4,  6, 12,  1;
  13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,  1;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
R. J. Mathar, Plots of cycle graphs of the finite groups up to order 36, (2015).
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A050873, A164306, A226314, A277227 (row reversed, k=0..n-1).

a054531 n k = div n $ gcd n k
a054531_row n = a054531_tabl !! (n-1)
a054531_tabl = zipWith (\u vs -> map (div u) vs) [1..] a050873_tabl
-- Reinhard Zumkeller, Jun 10 2013

Table[n/GCD[n,k], {n,1,10}, {k,1,n}]//Flatten (* G. C. Greubel, Sep 13 2017 *)

for(n=1,10, for(k=1,n, print1(n/gcd(n,k), ", "))) \\ G. C. Greubel, Sep 13 2017

A144437 Period 3: repeat [3, 3, 1].

Original entry on oeis.org

3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3

Offset: 1

Paul Curtz, Oct 05 2008

The sequence is generated from numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...
Conjecture: a(n) is the separatix. See A045944.
Also the decimal expansion of the constant 3310/999. - R. J. Mathar, May 21 2009
Continued fraction expansion of A171417.
Greatest common divisor of (n+1)^2-1 and (n+1)^2+2. - Bruno Berselli, Mar 08 2017

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000217, A045944, A109007, A164306, A167192, A169609, A256095.

Numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...

&cat [[3, 3, 1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

seq(op([3, 3, 1]), n=1..50); # Wesley Ivan Hurt, Jul 02 2016

A144437[n_]:=Denominator[n/3]; Array[A144437,100] (* Enrique Pérez Herrero, Oct 05 2011 *)
CoefficientList[Series[(3 + 3 x + x^2)/(1 - x^3), {x, 0, 120}], x] (* Michael De Vlieger, Jul 02 2016 *)
Table[Mod[2*n^2 + 1, 3,1], {n,1,50}] (* G. C. Greubel, Aug 24 2017 *)

a(n)=if(n%3,3,1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = (7-4*cos(2*Pi*n/3))/3. - Jaume Oliver Lafont, Nov 23 2008
G.f.: x*(3 + 3*x + x^2)/((1 - x)*(1 + x + x^2)). - R. J. Mathar, May 21 2009
a(n) = 3/gcd(n,3). - Reinhard Zumkeller, Oct 30 2009
a(n) = denominator(n^k/3), where k>0 is an integer. - Enrique Pérez Herrero, Oct 05 2011
a(n) = gcd(T(n+1), T(2)) = A256095(n+1, 2), with the triangular numbers T = A000217, for n >= 1. - Wolfdieter Lang, Mar 17 2015
a(n) = a(n-3) for n>3; a(n) = A169609(n) for n>0. - Wesley Ivan Hurt, Jul 02 2016
E.g.f.: (1/3)*(7*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2) - 3). - G. C. Greubel, Aug 24 2017
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 9/(a(n-2)*a(n-1)).
a(n) = 7 - a(n-2) - a(n-1). See also A052901 or A069705. (End)

Edited by R. J. Mathar, May 21 2009

A051537 Triangle read by rows: T(i,j) = lcm(i,j)/gcd(i,j) for 1 <= j <= i.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 2, 12, 1, 5, 10, 15, 20, 1, 6, 3, 2, 6, 30, 1, 7, 14, 21, 28, 35, 42, 1, 8, 4, 24, 2, 40, 12, 56, 1, 9, 18, 3, 36, 45, 6, 63, 72, 1, 10, 5, 30, 10, 2, 15, 70, 20, 90, 1, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1, 12, 6, 4, 3, 60, 2, 84, 6, 12, 30, 132, 1, 13, 26, 39

Offset: 1

N. J. A. Sloane and Amarnath Murthy, May 10 2002

From  Robert G. Wilson v, May 10 2002: (Start)
The first term of the k-th row is k. The first leading diagonal contains all 1's. The second leading diagonal contains twice the triangular numbers = n*(n-1).
For p prime, the sum of the p-th row is (p^3 - p^2 + 2)/2.
Proof: The p-th row is p, 2*p, 3*p, ..., (p-2)*p, (p-1)*p, 1. The sum of the row = p*(1 + 2 + 3 + ... + (p-2) + (p-1)) + 1 = p*(p-1)*p/2 + 1 = (p^3 - p^2 + 2)/2. (End) [Edited by Petros Hadjicostas, May 27 2020]
In the square array where T(i,j) = T(j,i), the natural extension of the triangle, each set of rows and columns with common indices [d1, d2, ..., ds] define a group multiplication table on their grid, if the d1, d2, ..., ds are the set of divisors of a squarefree number [A. Jorza]. - R. J. Mathar, May 03 2007
T(n,k) is the minimum number of squares necessary to fill a rectangle with sides of length n and k. - Stefano Spezia, Oct 06 2018

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins
  1;
  2,  1;
  3,  6,  1;
  4,  2, 12,  1;
  5, 10, 15, 20,  1;
  6,  3,  2,  6, 30,  1;
  7, 14, 21, 28, 35, 42,  1;
  8,  4, 24,  2, 40, 12, 56,  1;
  ...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Andrei Jorza, Groups of Divisors: Solution to problem 10893, Amer. Math. Monthly, 2003, 441-443.

Diagonals give A002378, A070260, A070261, A070262.

Row sums give A056789.

Flat(List([1..13],n->List([1..n],k->Lcm(n,k)/Gcd(n,k)))); # Muniru A Asiru, Oct 06 2018

a051537 n k = a051537_tabl !! (n-1) !! (k-1)
a051537_row n = a051537_tabl !! (n-1)
a051537_tabl = zipWith (zipWith div) a051173_tabl a050873_tabl
-- Reinhard Zumkeller, Jul 07 2013

/* As triangle */ [[Lcm(n,k)/Gcd(n,k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 07 2018

T:=proc(n,k) n*k/gcd(n,k)^2; end proc: seq(seq(T(n,k),k=1..n),n=1..13); # Muniru A Asiru, Oct 06 2018

Flatten[ Table[ LCM[i, j] / GCD[i, j], {i, 1, 13}, {j, 1, i}]]
T[n_,k_]:=n*k/GCD[n,k]^2; Flatten[Table[T[n,k],{k,1,13},{n,1,k}]] (* Stefano Spezia, Oct 06 2018 *)

T(n,k) = A054531(n,k)*A164306(n,k). - Reinhard Zumkeller, Oct 30 2009
T(n,k) = A051173(n,k) / A050873(n,k). - Reinhard Zumkeller, Jul 07 2013
T(n,k) = n*k/gcd(n,k)^2. - Stefano Spezia, Oct 06 2018

More terms from Robert G. Wilson v, May 10 2002

A167192 Triangle read by rows: T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 3, 2, 1, 0, 5, 2, 1, 1, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 3, 5, 1, 3, 1, 1, 0, 8, 7, 2, 5, 4, 1, 2, 1, 0, 9, 4, 7, 3, 1, 2, 3, 1, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 5, 3, 2, 7, 1, 5, 1, 1, 1, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 6, 11, 5, 9, 4, 1, 3, 5, 2, 3

Offset: 1

Reinhard Zumkeller, Oct 30 2009

			The triangle T(n,k) begins:
n\k   1   2   3   4  5  6  7  8  9 10 11 12 13  14  15 ...
1:    0
2:    1   0
3:    2   1   0
4:    3   1   1   0
5:    4   3   2   1  0
6:    5   2   1   1  1  0
7:    6   5   4   3  2  1  0
8:    7   3   5   1  3  1  1  0
9:    8   7   2   5  4  1  2  1  0
10:   9   4   7   3  1  2  3  1  1  0
11:  10   9   8   7  6  5  4  3  2  1  0
12:  11   5   3   2  7  1  5  1  1  1  1  0
13:  12  11  10   9  8  7  6  5  4  3  2  1  0
14:  13   6  11   5  9  4  1  3  5  2  3  1  1   0
15:  14  13   4  11  2  3  8  7  2  1  4  1  2   1   0
- _Wolfdieter Lang_, Feb 20 2013

Indranil Ghosh, Rows 1..120 of triangle, flattened

Cf. A054531, A112543, A164306.

Flatten[Table[(n-k)/GCD[n,k],{n,20},{k,n}]] (* Harvey P. Dale, Nov 27 2015 *)

for(n=1,10, for(k=1,n, print1((n-k)/gcd(n,k), ", "))) \\ G. C. Greubel, Sep 13 2017

T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.
T(n,k) = A025581(n,k)/A050873(n,k);
T(n,1) = A001477(n-1);
T(n,2) = A026741(n-2) for n > 1;
T(n,3) = A051176(n-3) for n > 2;
T(n,4) = A060819(n-4) for n > 4;
T(n,n-3) = A144437(n) for n > 3;
T(n,n-2) = A000034(n) for n > 2;
T(n,n-1) = A000012(n);
T(n,n) = A000004(n).

A343762 a(1) = 1; a(n) = -Sum_{k=1..n} a(k/gcd(n,k)).

Original entry on oeis.org

1, -2, 0, -3, 3, -5, 5, -12, 8, -12, 16, -31, 31, -55, 23, -99, 131, -184, 184, -389, 157, -528, 760, -1171, 800, -2058, 1235, -3248, 4442, -5566, 5566, -13461, 7433, -20534, 18290, -30439, 38711, -77429, 46895, -105973, 136507, -187059, 187059, -441337, 185384, -632122, 888075

Offset: 1

Ilya Gutkovskiy, Apr 28 2021

sign

Cf. A006512, A023900, A164306, A333613, A343761.

a[1] = 1; a[n_] := a[n] = -Sum[a[k/GCD[n, k]], {k, 1, n}]; Table[a[n], {n, 1, 47}]
a[1] = 1; a[n_] := a[n] = -Sum[Sum[If[GCD[k, d] == 1, a[k], 0], {k, 1, d}], {d, Divisors[n]}]; Table[a[n], {n, 1, 47}]

a(1) = 1; a(n) = -Sum_{d|n} Sum_{k=1..d, gcd(d,k) = 1} a(k).

a(n) = -a(n-1) if n belongs to A006512.

cp's OEIS Frontend

A054521 Triangle read by rows: T(n,k) = 1 if gcd(n, k) = 1, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Sage

Formula

A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

Python

Formula

Extensions

A054531 Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A144437 Period 3: repeat [3, 3, 1].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A051537 Triangle read by rows: T(i,j) = lcm(i,j)/gcd(i,j) for 1 <= j <= i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

Formula

Extensions

A167192 Triangle read by rows: T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.

Views

Author

Keywords

Examples