A054522 -id:A054522 - cp's OEIS frontend

A127478 Triangle T(n,k) read by rows: matrix product A054523 * A054522.

1, 2, 1, 3, 0, 2, 4, 2, 0, 2, 5, 0, 0, 0, 4, 6, 3, 4, 0, 0, 2, 7, 0, 0, 0, 0, 0, 6, 8, 4, 0, 4, 0, 0, 0, 4, 9, 0, 6, 0, 0, 0, 0, 0, 6, 10, 5, 0, 0, 8, 0, 0, 0, 0, 4, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 12, 6, 8, 6, 0, 4, 0, 0, 0, 0, 0, 4, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 14, 7, 0, 0, 0, 0, 12, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

If the two matrices A054523 and A054522 are commuted, the matrix product becomes A127477.

			First few rows of the triangle are:
.1;
.2, 1;
.3, 0, 2;
.4, 2, 0, 2;
.5, 0, 0, 0, 4;
.6, 3, 4, 0, 0, 2;
.7, 0, 0, 0, 0, 0, 6;
.8, 4, 0, 4, 0, 0, 0, 4;
....

Cf. A054522, A054523, A018804, A000010.

A054522 := proc(n,k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
A054523 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
A127478 := proc(n,k) add( A054523(n,j)*A054522(j,k), j=k..n) ; end: seq(seq( A127478(n,k),k=1..n),n=1..15) ;

T(n,k) = sum_{j=k..n} A054523(n,j) * A054522(j,k).
T(n,n) = A000010(n) (diagonal).
sum_{k=1..n} T(n,k) = A018804(n) (row sums).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

A130585 A054522 * A007318.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 7, 6, 2, 5, 16, 24, 16, 4, 6, 15, 22, 20, 10, 2, 7, 36, 90, 120, 90, 36, 6, 8, 35, 90, 142, 140, 84, 28, 4, 9, 52, 170, 336, 420, 336, 168, 48, 6, 10, 53, 168, 352, 508, 504, 336, 144, 36, 4

Offset: 0

Gary W. Adamson, Jun 06 2007

A130584 = A007318 * A054522 Row sums = A130586: (1, 3, 9, 19, 65, 75, 385, ...).

			First few rows of the triangle:
  1;
  2,  1;
  3,  4,  2;
  4,  7,  6,   2;
  5, 16, 24,  16,  4;
  6, 15, 22,  20, 10,  2;
  7, 36, 90, 120, 90, 36, 6;
  ...

Cf. A054522, A007318, A130585, A130586.

A054522 * A007318 as infinite lower triangular matrices.

A130584 A007318 * A054522.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 8, 4, 6, 2, 16, 8, 12, 8, 4, 32, 16, 22, 20, 20, 2, 64, 32, 42, 40, 60, 12, 6, 128, 64, 84, 72, 140, 42, 42, 4, 256, 128, 170, 128, 280, 112, 168, 32, 6, 512, 256, 342, 240, 508, 252, 504, 144, 54, 4

Offset: 0

Gary W. Adamson, Jun 06 2007

			First few rows of the triangle are:
1;
2, 1;
4, 2, 2;
8, 4, 6, 2;
16, 8, 12, 8, 4;
32, 16, 22, 20, 20, 2;
64, 32, 42, 40, 60, 12, 6;
...

Cf. A007318, A054522, A001792 (row sums), A130585, A000010.

T(n,n)=A000010(n+1).

Binomial transform of A054522 as infinite lower triangular matrices.

A127477 Triangle T(n,k) read by rows: matrix product A054522 * A054523.

Original entry on oeis.org

1, 2, 1, 5, 0, 2, 6, 3, 0, 2, 17, 0, 0, 0, 4, 10, 5, 4, 0, 0, 2, 37, 0, 0, 0, 0, 0, 6, 22, 11, 0, 6, 0, 0, 0, 4, 41, 0, 14, 0, 0, 0, 0, 0, 6, 34, 17, 0, 0, 8, 0, 0, 0, 0, 4, 101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 30, 15, 12, 10, 0, 6, 0, 0, 0, 0, 0, 4, 145, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 74, 37, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

If the two matrices A054522 and A054523 are commuted, the matrix product becomes A127478.

			First few rows of the triangle are:
1;
2, 1;
5, 0, 2;
6, 3, 0, 2;
17, 0, 0, 0, 4;
10, 5, 4, 0, 0, 2;
37, 0, 0, 0, 0, 0, 6;
22, 11, 0, 6, 0, 0, 0, 4;

Cf. A054522, A054523, A057660, A000010, A029939.

A054522 := proc(n,k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
A054523 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
A127477 := proc(n,k) add( A054522(n,j)*A054523(j,k), j=k..n) ; end: seq(seq( A127477(n,k),k=1..n),n=1..15) ;

T(n,k) = sum_{j=k..n} A054522(n,j) * A054523(j,k).
sum_{k=1..n} T(n,k) = A057660(n) (row sums).
T(n,n) = A000010(n) (diagonal).
T(n,1) = A029939(n).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

A130211 Triangle read by rows: matrix product A054522 * A000012.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, May 17 2007

			First few rows of the triangle are:
1;
2, 1;
3, 2, 2;
4, 3, 2, 2;
5, 4, 4, 4, 4;
6, 5, 4, 2, 2, 2;
7, 6, 6, 6, 6, 6, 6;
8, 7, 6, 6, 4, 4, 4, 4;
...

Cf. A000010, A054522, A130212 (product with swapped order), A057660 (row sums).

A130211 := proc(n,k)
    add( A054522(n,j),j=k..n) ;
end proc:
seq(seq(A130211(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 06 2016

A054522 * A000012 as infinite lower triangular matrices.

T(n,n) = A000010(n).

A127471 Triangle formed from the matrix product A051731 * A054522 of infinite lower triangular matrices, read by rows.

Original entry on oeis.org

1, 2, 1, 2, 0, 2, 3, 2, 0, 2, 2, 0, 0, 0, 4, 4, 2, 4, 0, 0, 2, 2, 0, 0, 0, 0, 0, 6, 4, 3, 0, 4, 0, 0, 0, 4, 3, 0, 4, 0, 0, 0, 0, 0, 6, 4, 2, 0, 0, 8, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 6, 4, 6, 4, 0, 4, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson, Jan 15 2007

Left column = (1, 2, 2, 3, 2, 4, ...) = d(n), A000005; right border = (1, 1, 2, 2, 4, 2, 6, ...) = phi(n), A000010; row sums = (1, 3, 4, 7, 6, 12, ...) = sigma(n), A000203.

			First few rows of the triangle:
  1;
  2, 1;
  2, 0, 2;
  3, 2, 0, 2;
  2, 0, 0, 0, 4;
  4, 2, 4, 0, 0, 2;
  2, 0, 0, 0, 0, 0, 6;
  4, 3, 0, 4, 0, 0, 0, 4;
  3, 0, 4, 0, 0, 0, 0, 0, 6;
  4, 2, 0, 0, 8, 0, 0, 0, 0, 4;
  ...

Cf. A000005, A000010, A000203, A051731, A054522.

Edited by N. J. A. Sloane, Aug 11 2019

a(49) = 0 inserted and more terms from Georg Fischer, May 29 2023

A127474 Triangle, square of A054522.

Original entry on oeis.org

1, 2, 1, 3, 0, 4, 4, 3, 0, 4, 5, 0, 0, 0, 16, 6, 3, 8, 0, 0, 4, 7, 0, 0, 0, 0, 0, 36, 8, 7, 0, 12, 0, 0, 0, 16, 9, 0, 16, 0, 0, 0, 0, 0, 36, 10, 5, 0, 0, 32, 0, 0, 0, 0, 16

Offset: 1

Gary W. Adamson, Jan 15 2007

Right border = A127473, squares of phi(n) terms. Row sums = A057660: (1, 3, 7, 11, 21, ...).

			First few rows of the triangle:
  1;
  2, 1;
  3, 0, 4;
  4, 3, 0,  4;
  5, 0, 0,  0, 16;
  6, 3, 8,  0,  0, 4;
  7, 0, 0,  0,  0, 0, 36;
  8, 7, 0, 12,  0, 0,  0, 16;
  ...

Cf. A127473, A057660, A054522.

(A054522)^2 as an infinite lower triangular matrix.

A127479 Triangle read by rows: A054522 * A054521 as infinite lower triangular matrices.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jan 15 2007

Row sums = A029939: (1, 2, 5, 6, 17, 10, 37, ...).

			First few rows of the triangle:
  1;
  2, 0;
  3, 2, 0;
  4, 0, 2, 0;
  5, 4, 4, 4, 0;
  6, 2, 0, 0, 2, 0;
  7, 6, 6, 6, 6, 6, 0;
  8, 0, 6, 0, 4, 0, 4, 0;
  ...

Cf. A029939, A054522, A054521.

Edited by N. J. A. Sloane, Aug 10 2019

A057660 a(n) = Sum_{k=1..n} n/gcd(n,k).

Original entry on oeis.org

1, 3, 7, 11, 21, 21, 43, 43, 61, 63, 111, 77, 157, 129, 147, 171, 273, 183, 343, 231, 301, 333, 507, 301, 521, 471, 547, 473, 813, 441, 931, 683, 777, 819, 903, 671, 1333, 1029, 1099, 903, 1641, 903, 1807, 1221, 1281, 1521, 2163, 1197, 2101, 1563, 1911, 1727

Offset: 1

Henry Gould, Oct 15 2000

Also sum of the orders of the elements in a cyclic group with n elements, i.e., row sums of A054531. - Avi Peretz (njk(AT)netvision.net.il), Mar 31 2001
Also inverse Moebius transform of EulerPhi(n^2), A002618.
Sequence is multiplicative with a(p^e) = (p^(2*e+1)+1)/(p+1). Example: a(10) = a(2)*a(5) = 3*21 = 63.
a(n) is the number of pairs (a, b) such that the equation ax = b is solvable in the ring (Zn, +, x). See the Mathematical Reflections link. - Michel Marcus, Jan 07 2017
From Jake Duzyk, Jun 06 2023: (Start)
These are the "contraharmonic means" of the improper divisors of square integers (inclusive of 1 and the square integer itself).
Permitting "Contraharmonic Divisor Numbers" to be defined analogously to Øystein Ore's Harmonic Divisor Numbers, the only numbers for which there exists an integer contraharmonic mean of the divisors are the square numbers, and a(n) is the n-th integer contraharmonic mean, expressible also as the sum of squares of divisors of n^2 divided by the sum of divisors of n^2. That is, a(n) = sigma_2(n^2)/sigma(n^2).
(a(n) = A001157(k)/A000203(k) where k is the n-th number such that A001157(k)/A000203(k) is an integer, i.e., k = n^2.)
This sequence is an analog of A001600 (Harmonic means of divisors of harmonic numbers) and A102187 (Arithmetic means of divisors of arithmetic numbers). (End)

David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 152.
H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), Vol. 39, No. 1 (1997), pp. 11-35.
H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), Vol. 39, No. 2 (1997), pp. 183-194.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Habib Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050)
Habib Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022)
Miriam Mahannah El-Farrah, Expectation Numbers of Cyclic Groups, MS Thesis, Western Kentucky University, August 2015.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 154-5.
Walther Janous, Problem 10829, Amer. Math. Monthly, 107 (2000), p. 753.
Yadollah Marefat, Ali Iranmanesh, and Abolfazl Tehranian, On the sum of element orders of finite simple groups, J. Algebra Applications, 12 (2013), #1350026.
Mathematical Reflections, Solution to Problem U338, Issue 4, 2015, p 17.
Joachim von zur Gathen, Arnold Knopfmacher, Florian Luca, Lutz G. Lucht, and Igor E. Shparlinski, Average order of cyclic groups, J. Théorie Nombres Bordeaux 16 (1) (2004) 107-123.

Cf. A000010, A000203, A000290, A001157, A018804, A050873, A051193, A054522, A057661, A061255, A065764, A078747, A174405 (partial sums), A176797, A226512.

Cf. A308471.

a057660 n = sum $ map (div n) $ a050873_row n
-- Reinhard Zumkeller, Nov 25 2013

Table[ DivisorSigma[ 2, n^2 ] / DivisorSigma[ 1, n^2 ], {n, 1, 128} ]
Table[Total[Denominator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 21 2020 *)

a(n)=if(n<1,0,sumdiv(n,d,d*eulerphi(d)))

a(n)=sumdivmult(n,d, eulerphi(d)*d) \\ Charles R Greathouse IV, Sep 09 2014

from math import gcd
def A057660(n): return sum(n//gcd(n,k) for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A057660(n): return prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items()) # Chai Wah Wu, Aug 05 2024

a(n) = Sum_{d|n} d*A000010(d) = Sum_{d|n} d*A054522(n,d), sum of d times phi(d) for all divisors d of n, where phi is Euler's phi function.
a(n) = sigma_2(n^2)/sigma_1(n^2) = A001157(A000290(n))/A000203(A000290(n)) = A001157(A000290(n))/A065764(n). - Labos Elemer, Nov 21 2001
a(n) = Sum_{d|n} A000010(d^2). - Enrique Pérez Herrero, Jul 12 2010
a(n) <= (n-1)*n + 1, with equality if and only if n is noncomposite. - Daniel Forgues, Apr 30 2013
G.f.: Sum_{n >= 1} n*phi(n)*x^n/(1 - x^n) = x + 3*x^2 + 7*x^3 + 11*x^4 + .... Dirichlet g.f.: sum {n >= 1} a(n)/n^s = zeta(s)*zeta(s-2)/zeta(s-1) for Re s > 3.  Cf. A078747 and A176797. - Peter Bala, Dec 30 2013
a(n) = Sum_{i=1..n} numerator(n/i). - Wesley Ivan Hurt, Feb 26 2017
L.g.f.: -log(Product_{k>=1} (1 - x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = Sum_{k=1..n} lcm(n,k)/k.
a(n) = Sum_{k=1..n} gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
From Vaclav Kotesovec, Jun 13 2021: (Start)
Sum_{k=1..n} a(k)/k ~ 3*zeta(3)*n^2/Pi^2.
Sum_{k=1..n} k^2/a(k) ~ A345294 * n.
Sum_{k=1..n} k*A000010(k)/a(k) ~ A345295 * n. (End)
Sum_{k=1..n} a(k) ~ 2*zeta(3)*n^3/Pi^2. - Vaclav Kotesovec, Jun 10 2023

More terms from James Sellers, Oct 16 2000

A003989 Triangle T from the array A(x, y) = gcd(x,y), for x >= 1, y >= 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 1, 2, 1

Offset: 1

Marc LeBrun

For m < n, the maximal number of nonattacking queens that can be placed on the n by m rectangular toroidal chessboard is gcd(m,n), except in the case m=3, n=6.
The determinant of the matrix of the first n rows and columns is A001088(n). [Smith, Mansion] - Michael Somos, Jun 25 2012
Imagine a torus having regular polygonal cross-section of m sides. Now, break the torus and twist the free ends, preserving rotational symmetry, then reattach the ends. Let n be the number of faces passed in twisting the torus before reattaching it. For example, if n = m, then the torus has exactly one full twist. Do this for arbitrary m and n (m > 1, n > 0). Now, count the independent, closed paths on the surface of the resulting torus, where a path is "closed" if and only if it returns to its starting point after a finite number of times around the surface of the torus. Conjecture: this number is always gcd(m,n). NOTE: This figure constitutes a group with m and n the binary arguments and gcd(m,n) the resulting value. Twisting in the reverse direction is the inverse operation, and breaking & reattaching in place is the identity operation. - Jason Richardson-White, May 06 2013
Regarded as a triangle, table of gcd(n - k +1, k) for 1 <= k <= n. - Franklin T. Adams-Watters, Oct 09 2014
The n-th row of the triangle is 1,...,1, if and only if, n + 1 is prime. - Alexandra Hercilia Pereira Silva, Oct 03 2020

			The array A begins:
  [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
  [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]
  [1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]
  [1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]
  [1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]
  [1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]
  [1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]
  [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]
  [1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]
  [1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]
  [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]
  [1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]
  ...
The triangle T begins:
  n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  1:  1
  2:  1  1
  3:  1  2  1
  4:  1  1  1  1
  5:  1  2  3  2  1
  6:  1  1  1  1  1  1
  7:  1  2  1  4  1  2  1
  8:  1  1  3  1  1  3  1  1
  9:  1  2  1  2  5  2  1  2  1
 10:  1  1  1  1  1  1  1  1  1  1
 11:  1  2  3  4  1  6  1  4  3  2  1
 12:  1  1  1  1  1  1  1  1  1  1  1  1
 13:  1  2  1  2  1  2  7  2  1  2  1  2  1
 14:  1  1  3  1  5  3  1  1  3  5  1  3  1  1
 15:  1  2  1  4  1  2  1  8  1  2  1  4  1  2  1
 ...  - _Wolfdieter Lang_, May 12 2018

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, ch. 4.
D. E. Knuth, The Art of Computer Programming, Addison-Wesley, section 4.5.2.

T. D. Noe, First 100 antidiagonals of array, flattened
Grant Cairns, Queens on Non-square Tori, El. J. Combinatorics, N6, 2001.
Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.
Warren P. Johnson, An LDU Factorization in Elementary Number Theory, Mathematics Magazine, Vol. 76, No. 5 (Dec., 2003), pp. 392-394.
P. Mansion, On an Arithmetical Theorem of Professor Smith's, Messenger of Mathematics, (1878), pp. 81-82.
Kival Ngaokrajang, Pattern of GCD(x,y) > 1 for x and y = 1..60. Non-isolated values larger than 1 (polyomino shapes) are colored.
Marcelo Polezzi, A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor, The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 445-446.
Mihai Prunescu and Joseph Shunia, Arithmetic-term representations for the greatest common divisor, arXiv:2411.06430 [math.NT], 2024.
H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.
Index entries for sequences related to lcm's

Rows, columns and diagonals: A089128, A109007, A109008, A109009, A109010, A109011, A109012, A109013, A109014, A109015.
A109004 is (0, 0) based.
Cf.  A001088, A003990, A003991, A050873, A051731, A054431, A054522.
Cf. also A091255 for GF(2)[X] polynomial analog.
A(x, y) = A075174(A004198(A075173(x), A075173(y))) = A075176(A004198(A075175(x), A075175(y))).
Antidiagonal sums are in A006579.

Flat(List([1..15],n->List([1..n],k->Gcd(n-k+1,k)))); # Muniru A Asiru, Aug 26 2018

a:=(n,k)->gcd(n-k+1,k): seq(seq(a(n,k),k=1..n),n=1..15); # Muniru A Asiru, Aug 26 2018

Table[ GCD[x - y + 1, y], {x, 1, 15}, {y, 1, x}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

{A(n, m) = gcd(n, m)}; /* Michael Somos, Jun 25 2012 */

Multiplicative in both parameters with a(p^e, m) = gcd(p^e, m). - David W. Wilson, Jun 12 2005
T(n, k) = A(n - k + 1, k) = gcd(n - k + 1, k), n >= 1, k = 1..n. See a comment above and the Mathematica program. - Wolfdieter Lang, May 12 2018
Dirichlet generating function: Sum_{n>=1} Sum_{k>=1} gcd(n, k)/n^s/k^c = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(s + c). - Mats Granvik, Feb 13 2021
The LU decomposition of this square array = A051731 * transpose(A054522) (see Johnson (2003) or Chamberland (2013), p. 1673). - Peter Bala, Oct 15 2023

cp's OEIS Frontend

A127478 Triangle T(n,k) read by rows: matrix product A054523 * A054522.

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A130585 A054522 * A007318.

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A130584 A007318 * A054522.

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A127477 Triangle T(n,k) read by rows: matrix product A054522 * A054523.

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A130211 Triangle read by rows: matrix product A054522 * A000012.

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A127471 Triangle formed from the matrix product A051731 * A054522 of infinite lower triangular matrices, read by rows.

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A127474 Triangle, square of A054522.

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A127479 Triangle read by rows: A054522 * A054521 as infinite lower triangular matrices.

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A057660 a(n) = Sum_{k=1..n} n/gcd(n,k).

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