A130209 -id:A130209 - cp's OEIS frontend

A130210 Triangle read by rows: matrix product A051731 * A130209.

1, 1, 2, 1, 0, 2, 1, 2, 0, 3, 1, 0, 0, 0, 2, 1, 2, 2, 0, 0, 4, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 2, 0, 0, 0, 0, 0, 3, 1, 2, 0, 0, 2, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 3, 0, 4, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2

Offset: 1

Gary W. Adamson, May 17 2007

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

Cf. A000005, A007425 (row sums).

A130210 := proc(n,k)
    add( A051731(n,j)*A130209(j,k),j=k..n) ;
end proc:
seq(seq(A130210(n,k),k=1..n),n=1..15) ;  # R. J. Mathar, Aug 06 2016

Inverse Moebius transform of A130209.

T(n,n) = A000005(n).

A143271 Triangle read by rows: A130209 * A000012 * A127648.

Original entry on oeis.org

1, 2, 4, 2, 4, 6, 3, 6, 9, 12, 2, 4, 6, 8, 10, 4, 8, 12, 16, 20, 24, 2, 4, 6, 8, 10, 12, 14, 4, 8, 12, 16, 20, 24, 28, 32, 3, 6, 9, 12, 15, 18, 21, 24, 27, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = A143272: (1, 6, 12, 30, 30, 84, 56, ...).

Left border = A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, ...).

			First few rows of the triangle =
  1;
  2, 4;
  2, 4,  6;
  3, 6,  9, 12;
  2, 4,  6,  8, 10;
  4, 8, 12, 16, 20, 24;
  2, 4,  6,  8, 10, 12, 14;
  ...
T(5,3) = 6 = 2*3 = d(5)*3.

Cf. A000005, A127648, A130209, A143272.

tabl(nn) = for (n=1, nn, for (k=1, n, print1(numdiv(n)*k, ", "))); \\ Michel Marcus, Jun 05 2023

T(n,k) = d(n)*k.

a(62) corrected by Georg Fischer, Jun 05 2023

A143273 Triangle read by rows: A127648 * A000012 * A130209.

Original entry on oeis.org

1, 2, 4, 3, 6, 6, 4, 8, 8, 12, 5, 10, 10, 15, 10, 6, 12, 12, 18, 12, 24, 7, 14, 14, 21, 14, 28, 14, 8, 16, 16, 24, 16, 32, 16, 32, 9, 18, 18, 27, 18, 36, 18, 36, 27, 10, 20, 20, 30, 20, 40, 20, 40, 30, 40, 11, 22, 22, 33, 22, 44, 22, 44, 33, 44, 22

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = A143274: (1, 6, 15, 32, 50, 84, ...).

			First few rows of the triangle =
  1;
  2,  4;
  3,  6,  6;
  4,  8,  8, 12;
  5, 10, 10, 15, 10;
  6, 12, 12, 18, 12, 24;
  7, 14, 14, 21, 14, 28, 14;
  8, 16, 16, 24, 16, 32, 16, 32;
  ...
T(6,3) = 12 = 6*d(3) = 6*2, where A000005 = d(k) = (1, 2, 3, 2, 2, 4, 2, 4, 3, ...).

Cf. A000005, A127648, A130209, A143274.

tabl(nn) = for (n=1, nn, for (k=1, n, print1(n*numdiv(k), ", "))); \\ Michel Marcus, Mar 19 2016

T(n,k) = n*d(k).

a(48) = 20 inserted by Georg Fischer, Jun 05 2023

A130277 Triangle read by rows: A130209 * A051731 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 2, 0, 2, 3, 3, 0, 3, 2, 0, 0, 0, 2, 4, 4, 4, 0, 0, 4, 2, 0, 0, 0, 0, 0, 2, 4, 4, 0, 4, 0, 0, 0, 4, 3, 0, 3, 0, 0, 0, 0, 0, 3, 4, 4, 0, 0, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 6, 6, 6, 0, 6, 0, 0, 0, 0, 0, 6, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Gary W. Adamson, May 18 2007

Row sums = A035116: (1, 4, 4, 9, 4, 16, 4, 16, ...).

Right and left borders = A000005, d(n): (1, 2, 2, 3, 2, 4, 2, ...).

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

Cf. A000005, A035116, A051731, A130209.

a(56) corrected and more terms from Georg Fischer, Jun 05 2023

A130207 Diagonalized matrix of A000010, Euler totient function phi.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, May 16 2007

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

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 rows of the triangle

Cf. A000010, A130208, A130209.

A130207 := proc(n,k)
    if k = n then
        numtheory[phi](n);
    else
        0;
    end if;
end proc:
seq(seq(A130207(n,k),k=1..n),n=1..15) ;

for(n=1,9,for(k=2,n,print1("0, "));print1(eulerphi(n)", ")) \\ Charles R Greathouse IV, Feb 19 2013

A130207(n) = if(ispolygonal(n,3), eulerphi((sqrtint(1+(n*8))-1)/2), 0); \\ Antti Karttunen, Jan 17 2025

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

T(n,k) = 0, if k <> n.

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

A130208 Diagonalized matrix of A000203, sigma(n).

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, May 16 2007

A130207 replaces sigma(n) with phi(n), A000010. A130209 replaces sigma(n) with d(n), A000005.

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

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 rows of the triangle

Cf. A000203, A130207, A130209.

for(n=1,9,for(k=2,n,print1("0, "));print1(sigma(n)", ")) \\ Charles R Greathouse IV, Feb 14 2013

A130208(n) = if(ispolygonal(n,3), sigma((sqrtint(1+(n*8))-1)/2), 0); \\ Antti Karttunen, Jan 17 2025

Infinite lower triangular matrix with A000203, sigma(n), in the main diagonal and the rest zeros.

Data section extended up to a(120) by Antti Karttunen, Jan 17 2025

A133698 Triangle, diagonal = A001227 with the rest zeros.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Sep 21 2007

Lower triangular part of an infinite matrix with A001227 (number of odd divisors of n) as the main diagonal, and the rest filled with zeros. - Redacted from the original formula given by the author. - Antti Karttunen, Jan 18 2025

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

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 rows of the triangle

Cf. A001227, A130209, A133699.

A000265(n) = (n>>valuation(n,2));
A133698(n) = if(ispolygonal(n,3), numdiv(A000265((sqrtint(1+(n*8))-1)/2)), 0); \\ Antti Karttunen, Jan 18 2025

Offset corrected from 0 to 1 and data section extended to a(105) by Antti Karttunen, Jan 18 2025

A143235 Triangle read by rows: T(n,k) = tau(n)*tau(k), the product of the number of divisors.

Original entry on oeis.org

1, 2, 4, 2, 4, 4, 3, 6, 6, 9, 2, 4, 4, 6, 4, 4, 8, 8, 12, 8, 16, 2, 4, 4, 6, 4, 8, 4, 4, 8, 8, 12, 8, 16, 8, 16, 3, 6, 6, 9, 6, 12, 6, 12, 9, 4, 8, 8, 12, 8, 16, 8, 16, 12, 16, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 6, 12, 12, 18, 12, 24, 12, 24, 18, 24, 12, 36, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4

Offset: 1

Gary W. Adamson, Aug 01 2008

The triangle can also be created by the triple matrix product A130209 * A000012 * A130209.

			First few rows of the triangle =
  1;
  2, 4;
  2, 4, 4;
  3, 6, 6,  9;
  2, 4, 4,  6, 4;
  4, 8, 8, 12, 8, 16;
  2, 4, 4,  6, 4,  8, 4;
  4, 8, 8, 12, 8, 16, 8, 16;
  3, 6, 6,  9, 6, 12, 6, 12, 9;
  ...
T(9,6) = 12 = d(9)*d(6) = 3*4.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000005, A035116 (right diagonal), A143236 (row sums).

A143235:= func< n,k | NumberOfDivisors(n)*NumberOfDivisors(k) >;
[A143235(n,k): k in [1..n], n in [1..14]]; // G. C. Greubel, Sep 12 2024

A143235[n_, k_]:= DivisorSigma[0, n]*DivisorSigma[0, k];
Table[A143235[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Sep 12 2024 *)

def A143235(n,k): return sigma(n,0)*sigma(k,0)
flatten([[A143235(n,k) for k in range(1,n+1)] for n in range(1,15)]) # G. C. Greubel, Sep 12 2024

T(n,k) = A000005(n)*A000005(k), for 1 <= k <= n, n >= 1.

Sum_{k=1..n} T(n, k) = A143236(n) (row sums).

cp's OEIS Frontend

A130210 Triangle read by rows: matrix product A051731 * A130209.

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A143271 Triangle read by rows: A130209 * A000012 * A127648.

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A143273 Triangle read by rows: A127648 * A000012 * A130209.

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A130277 Triangle read by rows: A130209 * A051731 as infinite lower triangular matrices.

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A130207 Diagonalized matrix of A000010, Euler totient function phi.

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A130208 Diagonalized matrix of A000203, sigma(n).

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A133698 Triangle, diagonal = A001227 with the rest zeros.

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A143235 Triangle read by rows: T(n,k) = tau(n)*tau(k), the product of the number of divisors.

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