A051731 -id:A051731 - cp's OEIS frontend

A128187 Matrix product A128174 * A051731 read by rows.

1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 0, 1, 0, 1, 3, 3, 1, 1, 0, 1, 4, 0, 1, 0, 1, 0, 1, 4, 4, 1, 2, 0, 1, 0, 1, 5, 0, 2, 0, 1, 0, 1, 0, 1, 5, 5, 1, 2, 1, 1, 0, 1, 0, 1, 6, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 6, 6, 2, 3, 1, 2, 0, 1, 0, 1, 0, 1, 7, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 2, 3, 1, 2, 1, 1

Offset: 1

Gary W. Adamson, Mar 07 2007

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

Cf. A128188 (row sums), A128174, A051731.

A128187 := proc(n,k)
    add( A128174(n,j)*A051731(j,k), j=k..n) ;
end proc: # R. J. Mathar, Apr 16 2013

A128174 * A051731 as infinite lower triangular matrices.

A129237 A051731 * A129234.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Apr 05 2007

Left border = Sigma(n), A000203: (1, 3, 4, 7, 6, 12, 8, ...). A129236 = Moebius transform of A129234.

			First few rows of the triangle:
   1;
   3,  2;
   4,  0, 3;
   7,  5, 0, 4;
   6,  0, 0, 0, 5;
  12,  6, 7, 0, 0, 6;
   8,  0, 0, 0, 0, 0, 7;
  15, 10, 0, 9, 0, 0, 0, 8;
  ...

Cf. A129234, A129235, A129236, A054525, A051731, A000203.

A051731 * A129234 as infinite lower triangular matrices.

A129353 A051731 * A115361.

Original entry on oeis.org

1, 2, 1, 1, 0, 1, 3, 2, 0, 1, 1, 0, 0, 0, 1, 2, 1, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 2, 3, 1, 0, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 10 2007

The inverse Moebius transform of the first column of A115361 which is A209229 gives the first column of this sequence.

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

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A001511.
Row sums are A129628 (inverse Moebius transform of A001511).
Cf. A051731, A115361.

A129353 := proc(n,k)
        add( A051731(n,j)*A115361(j-1,k-1),j=k..n) ;
end proc: # R. J. Mathar, Jul 14 2012

T[n_, k_] := If[Mod[n, k] != 0, 0, 1 + IntegerExponent[n/k, 2]];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 08 2020, from PARI *)

T(n, k)={if(n%k, 0, 1 + valuation(n/k,2))} \\ Andrew Howroyd, Aug 04 2018

T(n,k) = A001511(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 04 2018

A130093 A051731 * a lower triangular matrix with A036987 on the main diagonal and the rest zeros.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, May 06 2007

Right border = A036987, the Fredholm-Rueppel sequence, (1, 1, 0, 1, 0, 0, 0, 1, 0, ...). Row sums = the ruler function, A001511: (1, 2, 1, 3, 1, 2, 1, 4, ...).
A130093 also = A047999 (Sierpinski's gasket) * A036987 diagonalized, as infinite lower triangular matrices. - Gary W. Adamson, Oct 21 2009
Eigensequence of A130093 = A001511, (same sequence as row sums). - Gary W. Adamson, Oct 21 2009
Equals Sierpinski's gasket, A047999 * A036987 (diagonalized); as infinite lower triangular matrices. - Gary W. Adamson, Oct 26 2009

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

Cf. A001511, A051731, A036987.

Cf. A047999. - Gary W. Adamson, Oct 21 2009, Oct 26 2009

Inverse Moebius transform of a lower triangular matrix with A036987 (the Fredholm-Rueppel sequence) on the main diagonal and the rest zeros.

A134673 A051731 + A127448 - I where I is the Identity matrix (A023531).

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 05 2007

Row sums = A073757: (1, 2, 3, 4, 5, 5, 7, 7, 8, 7, ...).

			First few rows of the triangle:
  1;
  0,  2;
  0,  0,  3;
  1, -1,  0,  4;
  0,  0,  0,  0,  5;
  2, -1, -2,  0,  0,  6;
  0,  0,  0,  0,  0,  0,  7;
  1,  1,  0, -3,  0,  0,  0,  8;
  ... [Typo corrected by _N. J. A. Sloane_, May 22 2010]

Cf. A127448, A073757, A051731.

A134673(n,k) = if(n%k,0,if(k==n,n,k*moebius(n/k)+1)); /* Max Alekseyev, Jan 07 2015 */

a(n) = A051731(n) + A127448(n) - A023531(n).

T(n,k) = k*A008683(n/k) + 1 if k divides n and k < n, T(n,k)=n for k=n, and T(n,k)=0 otherwise. - Max Alekseyev, Jan 07 2015

More terms from Max Alekseyev, Apr 03 2022

A143112 A051731 * A032742 = sum of largest proper divisors of the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 5, 8, 2, 14, 2, 10, 8, 16, 2, 18, 2, 20, 10, 14, 2, 30, 7, 16, 14, 26, 2, 32, 2, 32, 14, 20, 10, 44, 2, 22, 16, 44, 2, 42, 2, 38, 26, 26, 2, 62, 9, 38, 20, 44, 2, 54, 14, 58, 22, 32, 2, 80, 2, 34, 34, 64, 16, 62, 2, 56, 26, 58, 2, 96, 2, 40, 38, 62, 14, 72, 2, 92

Offset: 1

Gary W. Adamson and Mats Granvik, Jul 25 2008

nonn

Inverse Möbius transform of A032742. - Antti Karttunen, Sep 25 2018

			a(12) = 14. The divisors of 12 are shown in row 12 of triangle A127093: (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12). The largest proper divisors of these terms are (1, 1, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6), sum = 14. Or, we can take row of triangle A051731: (1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1) dot (1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6) = (1 + 1 + 1 + 2 + 0 + 3 + 0 + 0 + 0 + 0 + 0 + 6) = 14, where A032742 = (1, 1, 1, 2, 1, 3, 1, 4, 3, 5,...).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A051731, A127093, A032742, A300236, A305807, A305808.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A143112(n) = sumdiv(n,d,A032742(d)); \\ Antti Karttunen, Sep 25 2018

A051731 * A032742, where A051731 = the inverse Mobius transform and A032742 = the largest proper divisors of n: (1, 1, 1, 3, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7,...).

a(n) = Sum_{d|n} A032742(d). - Antti Karttunen, Sep 25 2018

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

A168508 Triangle read by rows: A101688 * A051731.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 27 2009

More precisely, form the product of the lower triangular matrix T defined in A101688 and the lower triangular matrix T defined in A051731. - N. J. A. Sloane, Dec 05 2020
Row sums = A060831: (1, 2, 4, 5, 7, 9, 11, 12, 15,...).
A168509 = A051731 * A101688

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

Cf. A051731, A060831, A101688, A168509.

a(29) = 1 inserted and more terms from Georg Fischer, May 29 2023

A168509 Triangle read by rows, A051731 * A101688.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 27 2009

Row sums = A079247: (1, 2, 3, 4, 4, 7, 5, 8, 8, 10,...).

A168508 = A101688 * A051731

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

Cf. A051731, A101688, A079247, A168508

Triangle read by rows, inverse Mobius transform of A101688; where A051731 = the

inverse Mobius transform operator.

A176702 Triangle T(n,k) read by rows. A051731(n,k)-A051731(n,k+1).

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Gary W. Adamson, Apr 24 2010

Matrix inverse of A134541.

The subsequence of A180430 beginning from the second column, divided by 2, equals this sequence. [From Mats Granvik, Sep 04 2010]

			Triangle begins:
1
0...1
1..-1...1
0...1..-1...1
1...0...0..-1...1
0...0...1...0..-1...1
1...0...0...0...0..-1...1
0...1..-1...1...0...0..-1...1
1..-1...1...0...0...0...0..-1...1
0...1...0..-1...1...0...0...0..-1...1
1...0...0...0...0...0...0...0...0..-1...1
0...0...0...1..-1...1...0...0...0...0..-1...1

Cf. A176702*A000012=A051731. Row sums equals A000012.

A127470 Triangle equal to the matrix product A127466 * A051731.

Original entry on oeis.org

1, 4, 2, 9, 0, 6, 16, 12, 0, 8, 25, 0, 0, 0, 20, 36, 18, 24, 0, 0, 12, 49, 0, 0, 0, 0, 0, 42, 64, 56, 0, 48, 0, 0, 0, 32, 81, 0, 72, 0, 0, 0, 0, 0, 54, 100, 50, 0, 0, 80, 0, 0, 0, 0, 40, 121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 144, 108, 96, 72, 0, 72, 0, 0, 0, 0, 0, 48

Offset: 1

Gary W. Adamson, Jan 15 2007

			First few rows of the triangle are:
1;
4, 2;
9, 0, 6;
16, 12, 0, 8;
25, 0, 0, 0, 20;
36, 18, 24, 0, 0, 12;
49, 0, 0, 0, 0, 0, 42;
64, 56, 0, 48, 0, 0, 0, 32;
81, 0, 72, 0, 0, 0, 0, 0, 54;
...

Cf. A127466, A051731, A127469 (row sums), A061949.

A127470 := proc(n,k)
    add( A127466(n,j)*A051731(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 08 2013

T(n,1) = n^2.

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

Terms corrected by R. J. Mathar, Sep 08 2013

cp's OEIS Frontend

A128187 Matrix product A128174 * A051731 read by rows.

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A129237 A051731 * A129234.

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A129353 A051731 * A115361.

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A130093 A051731 * a lower triangular matrix with A036987 on the main diagonal and the rest zeros.

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A134673 A051731 + A127448 - I where I is the Identity matrix (A023531).

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A143112 A051731 * A032742 = sum of largest proper divisors of the divisors of n.

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A168508 Triangle read by rows: A101688 * A051731.

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A168509 Triangle read by rows, A051731 * A101688.

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A176702 Triangle T(n,k) read by rows. A051731(n,k)-A051731(n,k+1).

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