A054525 -id:A054525 - cp's OEIS frontend

A129236 A054525 * A129234.

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

Offset: 1

Gary W. Adamson, Apr 05 2007

Left border = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, ...). A129237 = inverse Moebius transform of A129234.

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

Cf. A129234, A129235, A129237.

A054525 * A129234 as infinite lower triangular matrices.

A134540 A054525 * A000012.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, -1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, -1, 0, 0, 0, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Oct 31 2007

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ...).

			First few rows of the triangle:
  1;
  0,  1;
  0,  1, 1;
  0,  0, 1, 1;
  0,  1, 1, 1, 1;
  0, -1, 0, 1, 1, 1;
  0,  1, 1, 1, 1, 1, 1;
  0,  0, 0, 0, 1, 1, 1, 1;
  0,  0, 0, 1, 1, 1, 1, 1, 1;
  0, -1, 0, 0, 0, 1, 1, 1, 1, 1;
  ...

Cf. A054525.

A054525 * A000012 as infinite lower triangular matrices.

Triangle read by rows, partial sums of A054525 (the Mobius transform) terms starting from the right.

A140581 Triangle read by rows, A054525 * A140256.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 0, 1, 0, 1, 4, 0, 0, 0, 1, -3, 2, 1, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, -5, 4, 0, 0, 1, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -3, 0, 20, 1, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -7, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson and Mats Granvik, May 17 2008

Row sums = A014963: (1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1,...).

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

Cf. A140256, A054525.

Mobius transform of triangle A140256 = A054525 * A140256; as infinite lower triangular matrices.

A127468 Triangle read by rows: matrix product A127466*A054525.

Original entry on oeis.org

1, 0, 2, -3, 0, 6, 0, -4, 0, 8, -15, 0, 0, 0, 20, 0, -6, 0, 0, 0, 12, -35, 0, 0, 0, 0, 0, 42, 0, -8, 0, -16, 0, 0, 0, 32, -9, 0, -36, 0, 0, 0, 0, 0, 54, 0, -30, 0, 0, 0, 0, 0, 0, 0, 40, -99, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 0, 12, 0, -24, 0, -24, 0, 0, 0, 0, 0, 48

Offset: 1

Gary W. Adamson, Jan 15 2007

The row sums are n, the index of the row.

			First few rows of the triangle are:
1;
0, 2;
-3, 0, 6;
0, -4, 0, 8;
-15, 0, 0, 0, 20;
0, -6, 0, 0, 0, 12;
-35, 0, 0, 0, 0, 0, 42;
...

Cf. A127466, A054525, A002618.

A127468 := proc(n,k)
    add( A127466(n,j)*A054525(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 06 2013

T(n,k) = sum_{j=k..n} A127466(n,j) * A054525(j,k).

T(n,n) = A002618(n).

A127569 Triangle read by rows: product of the Mobius matrix A054525 by the diagonal matrix diag(A000203(n)).

Original entry on oeis.org

1, -1, 3, -1, 0, 4, 0, -3, 0, 7, -1, 0, 0, 0, 6, 1, -3, -4, 0, 0, 12, -1, 0, 0, 0, 0, 0, 8, 0, 0, 0, -7, 0, 0, 0, 15, 0, 0, -4, 0, 0, 0, 0, 0, 13, 1, -3, 0, 0, -6, 0, 0, 0, 0, 18, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 3, 0, -7, 0, -12, 0, 0, 0, 0, 0, 28, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 1, -3, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 24, 1, 0, -4, 0, -6, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 19 2007

Left column = mu(n), A008683; right border = sigma(n), A000203; row sums = n.

The definition of Mobius transform is extended to matrices here in the sense of "left multiplication by the Mobius matrix A054525". - R. J. Mathar, Oct 02 2007

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

Cf. A054525, A000203, A008683.

A000203T := proc(n,k) if n = k then numtheory[sigma](n) ; else 0 ; fi ; end: A054525 := proc(n,k) if n < 1 or k > n or n mod k <> 0 then 0; else numtheory[mobius](n/k) ; fi ; end: A127569 := proc(n,k) add(A054525(n,i)*A000203T(i,k),i=1..n) ; end: for n from 1 to 15 do for k from 1 to n do printf("%a, ",A127569(n,k)) ; od: od: # R. J. Mathar, Oct 02 2007

T(n,k)=A054525(n,k)*A000203(k). - R. J. Mathar, Oct 02 2007

Missing comma corrected by Naruto Canada, Aug 26 2007

More terms from R. J. Mathar, Oct 02 2007

A127704 A054525 * A127701.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jan 24 2007

Moebius transform of A127701.

Row sums = A127705: (1, 2, 3, 2, 5, 1, 7, 4, 6, 3, ...)

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

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A054525, A127701, A127705.

A054525 := proc(n,k) if n>=1 and 1<=k and k <= n then if n mod k = 0 then numtheory[mobius](n/k) ; else 0 ; fi; else 0 ; fi; end: A127701 := proc(n,k) if n<1 or k<1 or k > n then 0 ; elif n = k then n; elif k+ 1 =n then 1; else 0 ; fi; end: A127704 := proc(n,k) add( A054525(n,i)*A127701(i,k),i=1..n) ; end: for n from 1 to 30 do for k from 1 to n do printf("%d,",A127704(n,k)) ; od: od: # R. J. Mathar, Jul 21 2009

More terms from R. J. Mathar, Jul 21 2009

A130027 A130026 * A054525.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 1, 2, 0, 1, 4, 0, 0, 0, 1, -2, 4, 3, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 1, 2, 0, 4, 0, 0, 0, 1, 2, 0, 6, 0, 0, 0, 0, 0, 1, -4, 8, 0, 0, 5, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, May 02 2007

Row sums = (1, 2, 3, ...). Left column = signed version of A070777.

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

Cf. A130027, A070777.

A130026 * A054525 as infinite lower triangular matrices.

A131088 2*A051731 - A054525 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 3, 0, 1, 2, 3, 0, 1, 3, 0, 0, 0, 1, 1, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 2, 2, 0, 3, 0, 0, 0, 1, 2, 0, 3, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 2, 3, 0, 3, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jun 14 2007

Row sums = A131089.

A131090: (1, 3, 3, 2, 3, 1, 3, 2, 2, 1, ...) in every column interspersed with (k-1) zeros.

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

Cf. A129979 (left border), A131089 (row sums), A051731, A054525.

T(n,k) = 2*!(n%k) - if (!(n % k), moebius(n/k), 0);
row(n) = vector(n, k, T(n,k));
lista(nn) = for (n=1, nn, v = row(n); for (k=1, #v, print1(v[k], ", "))); \\ Michel Marcus, Feb 26 2022

More terms from Michel Marcus, Feb 26 2022

A133732 A054525 * A000041.

Original entry on oeis.org

1, 0, 1, 2, 4, 5, 10, 12, 20, 25, 41, 47, 76, 90, 129, 161, 230, 270, 384, 458, 615, 750, 1001, 1187, 1570, 1881, 2414, 2907, 3717, 4400, 5603, 6666, 8306, 9912, 12295, 14537, 17976, 21252, 25937, 30683, 37337, 43861, 53173, 62467, 75020, 88132

Offset: 1

Gary W. Adamson, Sep 22 2007

nonn

A000041 = (1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...).

			a(4) = 2 = (0, -1, 0, 1) dot (1, 1, 2, 3) = (0, -1, 0, 3).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Mircea Merca and Maxie D. Schmidt, Generating Special Arithmetic Functions by Lambert Series Factorizations, arXiv:1706.00393 [math.NT], 2017. See Conjecture 3.1.
Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.

Cf. A054525.

read("transforms") : A000041 := proc(n) combinat[numbpart](n) ; end: a000041 := [seq(A000041(n),n=0..150)] ; a133732 := MOBIUS(a000041) ; # R. J. Mathar, Jan 19 2009
mob := (m,n) -> if irem(m,n) = 0 then numtheory:-mobius(m/n) else 0 fi:
A133732 := n -> add(mob(n,d)*combinat:-numbpart(d-1), d=1..n):
seq(A133732(n), n=1..46); # Peter Luschny, Jan 20 2018

a[n_] := DivisorSum[n, MoebiusMu[n/#]*PartitionsP[#-1]&];
Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Jan 20 2018 *)

Möbius transform of A000041, the partition numbers.

More terms from R. J. Mathar, Jan 19 2009

Offset set to 1 by Peter Luschny, Jan 20 2018

A137585 Triangle read by rows: A054525 * A026794.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 1, 0, 0, 1, 5, 1, 0, 0, 0, 1, 10, 2, 1, 0, 0, 0, 1, 12, 3, 1, 0, 0, 0, 0, 1, 20, 4, 1, 1, 0, 0, 0, 0, 1, 25, 5, 2, 1, 0, 0, 0, 0, 0, 1, 41, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 47, 10, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 76, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 27 2008

Row sums = A000837 starting (1, 1, 2, 3, 6, 7, 14, 17, ...).

			First few rows of the triangle:
   1;
   0, 1;
   1, 0, 1;
   2, 0, 0, 1;
   4, 1, 0, 0, 1;
   5, 1, 0, 0, 0, 1;
  10, 2, 1, 0, 0, 0, 1;
  12, 3, 1, 0, 0, 0, 0, 1;
  ...

Cf. A054525, A026794, A000837.

Mobius transform of A026794, the partition triangle.

cp's OEIS Frontend

A129236 A054525 * A129234.

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A134540 A054525 * A000012.

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A140581 Triangle read by rows, A054525 * A140256.

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A127468 Triangle read by rows: matrix product A127466*A054525.

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A127569 Triangle read by rows: product of the Mobius matrix A054525 by the diagonal matrix diag(A000203(n)).

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Maple

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A127704 A054525 * A127701.

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A130027 A130026 * A054525.

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A131088 2*A051731 - A054525 as infinite lower triangular matrices.

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PARI

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A133732 A054525 * A000041.

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