A126988 -id:A126988 - cp's OEIS frontend

A181116 Triangle T(n,k) read by rows. T(n,k) = A046643(A126988).

1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 5, 3, 0, 1, 0, 0, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 3, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Mats Granvik, Oct 04 2010

Numerator in fraction A181116/A181117. A051731=(A181116/A181117)*(A181116/A181117).

			Triangle starts:
1,
1,1,
1,0,1,
3,1,0,1,
1,0,0,0,1,
1,1,1,0,0,1,
1,0,0,0,0,0,1,
5,3,0,1,0,0,0,1,
3,0,1,0,0,0,0,0,1,
1,1,0,0,1,0,0,0,0,1,
1,0,0,0,0,0,0,0,0,0,1,
3,1,3,1,0,1,0,0,0,0,0,1,
1,0,0,0,0,0,0,0,0,0,0,0,1,
1,1,0,0,0,0,1,0,0,0,0,0,0,1,
1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,

Cf. A181117, A046643, A051731.

A181117 Triangle T(n,k) read by rows. T(n,k) = A046644(A126988).

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 8, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 16, 8, 0, 2, 0, 0, 0, 1, 8, 0, 2, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 16, 4, 8, 2, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Mats Granvik, Oct 04 2010

Denominator in fraction A181116/A181117.

			Triangle starts:
1,
2,1,
2,0,1,
8,2,0,1,
2,0,0,0,1,
4,2,2,0,0,1,
2,0,0,0,0,0,1,
16,8,0,2,0,0,0,1,
8,0,2,0,0,0,0,0,1,
4,2,0,0,2,0,0,0,0,1,
2,0,0,0,0,0,0,0,0,0,1,
16,4,8,2,0,2,0,0,0,0,0,1,
2,0,0,0,0,0,0,0,0,0,0,0,1,
4,2,0,0,0,0,2,0,0,0,0,0,0,1,
4,0,2,0,2,0,0,0,0,0,0,0,0,0,1,

Cf. A181116, A046644, A051731.

A127168 Triangle read by rows: square of A126988.

Original entry on oeis.org

1, 4, 1, 6, 0, 1, 12, 4, 0, 1, 10, 0, 0, 0, 1, 24, 6, 4, 0, 0, 1, 14, 0, 0, 0, 0, 0, 1, 32, 12, 0, 4, 0, 0, 0, 1, 27, 0, 6, 0, 0, 0, 0, 0, 1, 40, 10, 0, 0, 4, 0, 0, 0, 0, 1, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 72, 24, 12, 6, 0, 4, 0, 0, 0, 0, 0, 1, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Gary W. Adamson, Jan 06 2007

Row sums = A060640: (1, 5, 7, 17, 11, 35, 15, 49...) Left column = A038040, d(n)*n: (1, 4, 6, 12, 10, 24, 14, 32, 27, 40...). A127168 * A008683 = A018804, (1, 3, 5, 8, 9, 15...); where A008683 = the Mobius sequence.

			First few rows of the triangle are:
1;
4, 1;
6, 0, 1;
12, 4, 0, 1
10, 0, 0, 0, 1
24, 6, 4, 0, 0, 1
14, 0, 0, 0, 0, 0, 1;
32, 12, 0, 4, 0, 0, 0, 1;
...

Cf. A008683, A018804, A038040, A060640, A127169.

a(20) = 1 inserted and more terms from Georg Fischer, May 31 2023

A127169 Triangle read by rows: cube of A126988.

Original entry on oeis.org

1, 6, 1, 9, 0, 1, 24, 6, 0, 1, 15, 0, 0, 0, 1, 54, 9, 6, 0, 0, 1, 21, 0, 0, 0, 0, 0, 1, 80, 24, 0, 6, 0, 0, 0, 1, 54, 0, 9, 0, 0, 0, 0, 0, 1, 90, 15, 0, 0, 6, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Left column = A034718: (1, 6, 9, 24, 15, 54, 21...)

			First few rows of the triangle are:
[0] 1;
[1] 6, 1;
[2] 9, 0, 1;
[3] 24, 6, 0, 1;
[4] 15, 0, 0, 0, 1;
[5] 54, 9, 6, 0, 0, 1;
[6] 21, 0, 0, 0, 0, 0, 1;
[7] 80, 24, 0, 6, 0, 0, 0, 1;
[8] 54, 0, 9, 0, 0, 0, 0, 0, 1;
[9] 90, 15, 0, 0, 6, 0, 0, 0, 0, 1;
...

Cf. A034718, A126988.

a(45), a(54) deleted by Georg Fischer, Jun 01 2023

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

A127445 Triangle defined by the matrix product A126988 * A127368, read by rows.

Original entry on oeis.org

1, 3, 0, 4, 2, 0, 7, 0, 3, 0, 6, 2, 3, 4, 0, 12, 4, 0, 0, 5, 0, 8, 2, 3, 4, 5, 6, 0, 15, 0, 9, 0, 5, 0, 7, 0, 13, 8, 0, 4, 5, 0, 7, 8, 0, 18, 4, 9, 8, 0, 0, 7, 0, 9, 0, 12, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 28, 8, 9, 0, 15, 0, 7, 0, 0, 0, 11

Offset: 1

Gary W. Adamson, Jan 14 2007

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

Cf. A126988, A127368, A000217 (row sums), A000203 (column k=1).

A127368:=proc(n, k) if igcd(n,k) = 1 then k else 0 fi end:
A127445 := proc(n,k)
        add (A126988(n,j)*A127368(j,k),j=k..n) ;
end proc:
seq(seq(A127445(n,k),k=1..n),n=1..14) ; # R. J. Mathar, Nov 11 2011

T(n,k) = Sum_{j=k..n} A126988(n,j) * A127368(j,k), 1<=k<=n.

A134839 Triangle, n-th row = first n terms in n-th row of an array formed by A051731 * A126988(transform).

Original entry on oeis.org

1, 1, 3, 1, 2, 4, 1, 3, 3, 7, 1, 2, 3, 4, 6, 1, 3, 4, 6, 5, 12, 1, 2, 3, 4, 5, 6, 8, 1, 3, 3, 7, 5, 9, 7, 15, 1, 2, 4, 4, 5, 8, 7, 8, 13, 1, 3, 3, 6, 6, 9, 7, 12, 9, 18

Offset: 1

Gary W. Adamson, Nov 12 2007

Right border = sigma(n), A000203: (1, 3, 4, 7, 6, 12, ...).

Row sums = A007437: (1, 4, 7, 14, 16, 31, 29, 50, ...).

			First few rows of the array:
  1,  2,  3,  4,  5,  6,  7, ...
  1,  3,  3,  6,  5,  9,  7, ...
  1,  2,  4,  4,  5,  8,  7, ...
  1,  3,  3,  7,  5,  9,  7, ...
  1,  2,  3,  4,  6,  6,  7, ...
  1,  3,  4,  6,  5, 12,  7, ...
  ...
First few rows of the triangle:
  1;
  1,  3;
  1,  2,  4;
  1,  3,  3,  7;
  1,  2,  3,  4,  6;
  1,  3,  4,  6,  5, 12;
  1,  2,  3,  4,  5,  6,  8;
  ...

Cf. A126988, A051731, A000203, A007437.

Given the array formed by A051731 * A126988(transform), the triangle by rows = first n terms of n-th row of the array.

A134840 Triangle, antidiagonals of an array formed by A051731 * A126988(transform).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 2, 3, 4, 1, 3, 4, 6, 5, 1, 2, 3, 4, 5, 6, 1, 3, 3, 7, 5, 9, 7, 1, 2, 4, 4, 5, 8, 7, 8, 1, 3, 3, 6, 6, 9, 7, 12, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 3, 4, 7, 5, 12, 7, 14, 12, 15, 11

Offset: 1

Gary W. Adamson, Nov 12 2007

Row sums = A069153: (1, 3, 7, 10, 19, 21, 35, 39, ...).

			First few rows of the array:
  1, 2, 3, 4, 5, 6, 7, ...
  1, 3, 3, 6, 5, 9, 7, ...
  1, 2, 4, 4, 5, 8, 7, ...
  1, 3, 3, 7, 5, 9, 7, ...
  1, 2, 3, 4, 6, 6, 7, ...
  ...
First few rows of the triangle:
  1;
  1, 2;
  1, 3, 3;
  1, 2, 3, 4;
  1, 3, 4, 6, 5;
  1, 2, 3, 4, 5, 6;
  1, 2, 3, 7, 5, 9, 7;
  1, 2, 4, 4, 5, 8, 7, 8;
  ...

Cf. A051731, A126988, A069153.

Triangle, antidiagonals of an array formed by A051731 * A126988(transform).

A134841 Triangle read by rows: first n terms of n-th row of an array formed by A126988 * A053731(transform).

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 6, 4, 7, 5, 5, 5, 5, 6, 6, 9, 8, 9, 6, 12, 7, 7, 7, 7, 7, 7, 8, 8, 12, 8, 14, 8, 12, 8, 15, 9, 9, 12, 9, 9, 12, 9, 9, 13, 10, 15, 10, 15, 12, 15, 10, 15, 10, 18

Offset: 0

Gary W. Adamson, Nov 12 2007

Right border = sigma(n), A000203: (1, 3, 4, 7, 6, 12, ...).

Row sums = A001157: (1, 5, 10, 21, 26, 50, 50, ...).

			First few terms of the array:
  1,  1,  1,  1,  1,  1,  1, ...
  2,  3,  2,  3,  2,  3,  2, ...
  3,  3,  4,  3,  3,  4,  3, ...
  4,  6,  4,  7,  4,  6,  4, ...
  5,  5,  5,  5,  5,  6,  5, ...
  6,  9,  8,  9,  6, 12,  6, ...
  7,  7,  7,  7,  7,  7,  8, ...
  ...
First few rows of the triangle:
  1;
  2,  3;
  3,  3,  4;
  4,  6,  4,  7;
  5,  5,  5,  5,  6;
  6,  9,  8,  9,  6, 12;
  7,  7,  7,  7,  7,  7,  8,
  ...

Cf. A051731, A126988, A001157, A000203.

A141674 Triangle T(n,m) read by rows: T(n,m) = sigma_0(n) * A126988(n,m).

Original entry on oeis.org

1, 4, 2, 6, 0, 2, 12, 6, 0, 3, 10, 0, 0, 0, 2, 24, 12, 8, 0, 0, 4, 14, 0, 0, 0, 0, 0, 2, 32, 16, 0, 8, 0, 0, 0, 4, 27, 0, 9, 0, 0, 0, 0, 0, 3, 40, 20, 0, 0, 8, 0, 0, 0, 0, 4, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 72, 36, 24, 18, 0, 12, 0, 0, 0, 0, 0, 6, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 56, 28, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 06 2008

Row sums are {1, 6, 8, 21, 12, 48, 16, 60, 39, 72}.

			Triangle begins
  { 1},
  { 4,  2},
  { 6,  0,  2},
  {12,  6,  0,  3},
  {10,  0,  0,  0,  2},
  {24, 12,  8,  0,  0,  4},
  {14,  0,  0,  0,  0,  0,  2},
  {32, 16,  0,  8,  0,  0,  0,  4},
  {27,  0,  9,  0,  0,  0,  0,  0,  3},
  {40, 20,  0,  0,  8,  0,  0,  0,  0,  4}

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A126988.

t[n_, m_] = DivisorSigma[0, n]*If[m == 1, n, If[Mod[n, m] == 0, n/m, 0]]; Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[%]

T(n,m) = sigma_0(n) * A126988(n,m).

T(n,m) = sigma_0(n) * if(m = 1, n, if(n mod m = 0, n/m, 0)).

Edited by the Associate Editors of the OEIS, Jun 10 2018

cp's OEIS Frontend

A181116 Triangle T(n,k) read by rows. T(n,k) = A046643(A126988).

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A181117 Triangle T(n,k) read by rows. T(n,k) = A046644(A126988).

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A127168 Triangle read by rows: square of A126988.

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A127169 Triangle read by rows: cube of A126988.

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A127332 A126988 * A002321.

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A127445 Triangle defined by the matrix product A126988 * A127368, read by rows.

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A134839 Triangle, n-th row = first n terms in n-th row of an array formed by A051731 * A126988(transform).

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A134840 Triangle, antidiagonals of an array formed by A051731 * A126988(transform).

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A134841 Triangle read by rows: first n terms of n-th row of an array formed by A126988 * A053731(transform).

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A141674 Triangle T(n,m) read by rows: T(n,m) = sigma_0(n) * A126988(n,m).

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