A051731 -id:A051731 - cp's OEIS frontend

A128260 A128229 * A051731.

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

Offset: 1

Gary W. Adamson, Feb 21 2007

Row sums = A128261: (1, 3, 6, 9, 14, 14, 26, 18, 35, 31, ...).

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

Cf. A128229, A051731, A128261, A128259.

A128382 Inverse Moebius transform operation performed 24 times on A000594: A051731^24 * A000594.

Original entry on oeis.org

1, 0, 276, -1748, 4854, 0, -16720, 44552, -107295, 0, 534636, -482448, -577714, 0, 1339704, 2528206, -6905910, 0, 10661444, -8484792, -4614720, 0, 18643296, 12296352, -25383005, 0, -75928312, 29226560, 128406654, 0, -52843144, -151821160, 147559536, 0, -81158880

Offset: 1

Gary W. Adamson, Feb 28 2007

Conjecture: given the inverse Moebius transform operation performed any k times (k=1,2,3,...); k=24 is the only such sequence with zeros. A weaker conjecture: "zero" occurs an infinite number of times in A128382.

Multiplicative because A000594 is. Each application of A051731 corresponds to an inverse Moebius transform. - Andrew Howroyd, Aug 03 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A051731, A000594, A128378, A128379, A128380, A128381.

nmax = 40;
M = Table[If[Mod[n, k] == 0, 1, 0], {n, nmax}, {k, nmax}];
MatrixPower[M, 24].RamanujanTau[Range[nmax]] (* Jean-François Alcover, Sep 20 2019 *)

seq(n, k=24)={my(u=vector(n,n,1), v=vector(n,n,ramanujantau(n))); for(i=1, k, v=dirmul(u,v)); v} \\ Andrew Howroyd, Aug 03 2018

Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = (Sum_{n>=1} A000594(n)/n^s)*zeta(s)^24. - Jianing Song, Aug 04 2018

Terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A130162 Triangle read by rows: A051731 * A000837 as a diagonalized matrix.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 0, 0, 0, 6, 1, 1, 2, 0, 0, 7, 1, 0, 0, 0, 0, 0, 14, 1, 1, 0, 3, 0, 0, 0, 17, 1, 0, 2, 0, 0, 0, 0, 0, 27, 1, 1, 0, 0, 6, 0, 0, 0, 0, 34, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 1, 1, 2, 3, 0, 7, 0, 0, 0, 0, 0, 63

Offset: 1

Gary W. Adamson, May 13 2007

Right border = A000837 (offset 1).

Row sums = partition numbers A000041 starting (1, 2, 3, 5, 7, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  0,  2;
  1,  1,  0,  3;
  1,  0,  0,  0,  6;
  1,  1,  2,  0,  0,  7;
  1,  0,  0,  0,  0,  0, 14;
  1,  1,  0,  3,  0,  0,  0, 17;
  ...

Cf. A000041, A000837, A051731.

rows = 12; A000837[n_] := Sum[ MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]; A000837diag = DiagonalMatrix[Array[A000837, rows]]; A051731 = Table[ If[Mod[n, k] == 0, 1, 0], {n, 1, rows}, {k, 1, rows}]; A130162 = A051731.A000837diag; Table[ A130162[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2013 *)

A051731 * A000837 (starting at offset 1) as a diagonalized matrix M, where M = T(n,k) = A000837(n) * 0^(n-k), 1<=k<=n; i.e., (1; 0,1; 0,0,2; 0,0,0,3; 0,0,0,0,6;...).

A051731 = inverse Moebius transform.

More terms from Jean-François Alcover, Oct 03 2013

Offset changed to 1 by Georg Fischer, Jun 27 2023

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

A133699 A051731 * A133698.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Sep 21 2007

Right border = A001227, the number of odd divisors of n.

Row sums = A133700: (1, 2, 3, 3, 3, 6, 3, 4, 6, 6, ...)

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

Cf. A051731, A133698, A133700.

Inverse Mobius transform of A133698, as infinite lower triangular matrices.

A133701 A133698 * A051731.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Sep 21 2007

Left and right borders = A001227: (1, 1, 2, 1, 2, 2, 2, ...), the number of odd divisors of n.

Row sums = A133702, the inverse Mobius transform of A023136.

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

Cf. A051731, A133698, A001227, A133702, A023136.

A133698 * A051731 as infinite lower triangular matrices.

A133702 A051731 * A023136.

Original entry on oeis.org

1, 2, 4, 3, 4, 8, 4, 4, 9, 8, 4, 12, 4, 8, 16, 5, 6, 18, 4, 12, 16, 8, 4, 16, 9, 8, 16, 12, 4, 32, 8, 6, 16, 12, 16, 27, 4, 8, 16, 16, 6, 32, 8, 12, 36, 8, 4, 20, 9, 18, 24, 12, 4, 32, 16, 16, 16, 8, 4, 48, 4, 16, 44, 7, 20, 32, 4, 18, 16, 32, 4, 36, 10, 8, 36, 12, 16, 32, 4, 20, 25, 12, 4, 48

Offset: 1

Gary W. Adamson, Sep 21 2007

nonn

A023136 = (1, 1, 3, 1, 3, 3, 3, 1, 5, 3, ...).

			a(4) = 3 = (1, 1, 0, 1) * (1, 1, 3, 1) = (1, 1, 0, 1), row 4 of triangle A133701.

Cf. A051731, A133698, A023136.

Inverse Mobius transform of A023136.

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

A134699 Triangle read by rows: A051731^2 * A000012.

Original entry on oeis.org

1, 3, 1, 3, 1, 1, 6, 3, 1, 1, 3, 1, 1, 1, 1, 9, 5, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 10, 6, 3, 3, 1, 1, 1, 1, 6, 3, 3, 1, 1, 1, 1, 1, 1, 9, 5, 3, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 18, 12, 8, 5, 3, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 06 2007

Left column = A007425.
Row sums = A007429: (1, 4, 5, 11, 7, 20, ...).
A134700 = A000012 * A127170.

			First few rows of the triangle:
   1;
   3, 1;
   3, 1, 1;
   6, 3, 1, 1;
   3, 1, 1, 1, 1;
   9, 5, 3, 1, 1, 1;
   3, 1, 1, 1, 1, 1, 1;
  10, 6, 3, 3, 1, 1, 1, 1;
  ...

Cf. A007425, A007429, A051731, A127170, A134700.

A051731^2 * A000012 = A127170 * A000012, as infinite lower triangular matrices.

More terms from Jinyuan Wang, Apr 29 2025

A137587 Triangle read by rows: A051731 * A026794.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 5, 2, 0, 1, 6, 1, 0, 0, 1, 11, 3, 2, 0, 0, 1, 12, 2, 1, 0, 0, 0, 1, 20, 6, 1, 2, 0, 0, 0, 1, 25, 4, 3, 1, 0, 0, 0, 0, 1, 37, 9, 2, 1, 2, 0, 0, 0, 0, 1, 43, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 70, 16, 6, 3, 1, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 27 2008

That is, regard A051731 and A026794 as lower triangular square matrices and multiply them, then take the lower triangle of the product,
Left column = A083710 starting (1, 2, 3, 5, 6, 11, 12, ...).
Row sums = A047968.

			First few rows of the triangle:
   1;
   2, 1;
   3, 0, 1;
   5, 2, 0, 1;
   6, 1, 0, 0, 1;
  11, 3, 2, 0, 0, 1;
  12, 2, 1, 0, 0, 0, 1;
  20, 6, 1, 2, 0, 0, 0, 1;
  25, 4, 3, 1, 0, 0, 0, 0, 1;
  ...

Cf. A026794, A051731, A083710, A047968.

Inverse mobius transform of the partition triangle, A026794.

Typo in 9th row corrected by M. F. Hasler, Jun 08 2009

A140705 A000012 * A051731^4.

Original entry on oeis.org

1, 5, 1, 9, 1, 1, 19, 5, 1, 1, 23, 5, 1, 1, 1, 39, 9, 5, 1, 1, 1, 43, 9, 5, 1, 1, 1, 1, 63, 19, 5, 5, 1, 1, 1, 1, 73, 19, 9, 5, 1, 1, 1, 1, 1, 89, 23, 9, 5, 5, 1, 1, 1, 1, 1, 93, 23, 9, 5, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, May 24 2008

Row sums = A061203: (1, 6, 11, 26, 31, 56,...).

Left column = A061202: (1, 5, 9, 19, 23, 39,...).

			First few rows of the triangle are:
1;
5, 1;
9, 1, 1;
19, 5, 1, 1;
23, 5, 1, 1, 1;
39, 9, 5, 1, 1, 1;
43, 9, 5, 1, 1, 1, 1;
63, 19, 5, 5, 1, 1, 1, 1;
...

Cf. A051731, A061203, A061202.

A000012 * A051731^4 as infinite lower triangular matrices, where A051731 = the inverse Mobius transform and A000012 = an infinite lower triangular matrix with all 1's.

a(60) split into 5,1 by Georg Fischer, Aug 27 2023

cp's OEIS Frontend

A128260 A128229 * A051731.

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A128382 Inverse Moebius transform operation performed 24 times on A000594: A051731^24 * A000594.

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A130162 Triangle read by rows: A051731 * A000837 as a diagonalized matrix.

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

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A133699 A051731 * A133698.

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A133701 A133698 * A051731.

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A133702 A051731 * A023136.

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A134699 Triangle read by rows: A051731^2 * A000012.

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A137587 Triangle read by rows: A051731 * A026794.

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