cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Gary W. Adamson, May 16 2007

Keywords

Examples

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

Links

Crossrefs

Cf. A000010, A130208, A130209.

Programs

  • Maple
    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) ;
  • PARI
    for(n=1,9,for(k=2,n,print1("0, "));print1(eulerphi(n)", ")) \\ Charles R Greathouse IV, Feb 19 2013
  • PARI
    A130207(n) = if(ispolygonal(n,3), eulerphi((sqrtint(1+(n*8))-1)/2), 0); \\ Antti Karttunen, Jan 17 2025

Formula

T(n,n) = A000010(n).
T(n,k) = 0, if k <> n.

Extensions

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

A143237 Triangle read by rows, T(n, k) = A000203(n)*A000203(k), for n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 3, 9, 4, 12, 16, 7, 21, 28, 49, 6, 18, 24, 42, 36, 12, 36, 48, 84, 72, 144, 8, 24, 32, 56, 48, 96, 64, 15, 45, 60, 105, 90, 180, 120, 225, 13, 39, 52, 91, 78, 156, 104, 195, 169, 18, 54, 72, 126, 108, 216, 144, 270, 234, 324, 12, 36, 48, 84, 72, 144, 96, 180, 156, 216, 144
Offset: 1

Views

Author

Gary W. Adamson, Aug 01 2008

Keywords

Examples

			First few rows of the triangle =
   1;
   3,  9;
   4, 12, 16;
   7, 21, 28,  49;
   6, 18, 24,  42, 36;
  12, 36, 48,  84, 72, 144;
   8, 24, 32,  56, 48,  96,  64;
  15, 45, 60, 105, 90, 180, 120, 225;
  13, 39, 52,  91, 78, 156, 104, 195, 169;
  ...
T(6,3) = 48 = sigma(6)*sigma(3) = 12*4

Links

Crossrefs

Cf. A000203, A024916, A072861 (right diagonal), A130208, A143238 (row sums).

Programs

  • Magma
    A143237:= func< n,k | DivisorSigma(1,n)*DivisorSigma(1,k) >;
[A143237(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Sep 12 2024
  • Mathematica
    A143237[n_, k_]:= DivisorSigma[1,n]*DivisorSigma[1,k];
Table[A143237[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Sep 12 2024 *)
  • SageMath
    def A143237(n,k): return sigma(n,1)*sigma(k,1)
flatten([[A143237(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Sep 12 2024

Formula

Triangle read by rows, A130208 * A000012 * A130208, for 1 <= k <= n, n >= 1.
T(n, k) = sigma(n)*sigma(k), where sigma(n) = A000203(n).
Sum_{k=1..n} T(n, k) = A143238(n) (row sums).

Extensions

New title by G. C. Greubel, Sep 12 2024

A127574 Triangle T(n,k) = k*sigma(n) if k divides n, else 0.

Original entry on oeis.org

1, 3, 6, 4, 0, 12, 7, 14, 0, 28, 6, 0, 0, 0, 30, 12, 24, 36, 0, 0, 72, 8, 0, 0, 0, 0, 0, 56, 15, 30, 0, 60, 0, 0, 0, 120, 13, 0, 39, 0, 0, 0, 0, 0, 117, 18, 36, 0, 0, 90, 0, 0, 0, 0, 180, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 132, 28, 56, 84, 112, 0, 168, 0, 0, 0, 0, 0, 336
Offset: 1

Views

Author

Gary W. Adamson, Jan 19 2007

Keywords

Examples

			First few rows of the triangle are:
   1;
   3,  6;
   4,  0, 12;
   7, 14,  0, 28;
   6,  0,  0,  0, 30;
  12, 24, 36,  0,  0, 72;
  ...

Crossrefs

Cf. A127093, A127573, A064987, A000203, A072861 (row sums).

Formula

T(n,k) = Sum_{j=k..n} A130208(n,j)*A127093(j,k), product of the two infinite lower triangular matrices.
T(n,1) = A000203(n).
T(n,n) = A064987(n).

A127570 Triangle T(n,k) = sigma(k) if k|n, otherwise T(n,k)=0; 1 <= k <= n.

Original entry on oeis.org

1, 1, 3, 1, 0, 4, 1, 3, 0, 7, 1, 0, 0, 0, 6, 1, 3, 4, 0, 0, 12, 1, 0, 0, 0, 0, 0, 8, 1, 3, 0, 7, 0, 0, 0, 15, 1, 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, 1, 3, 4, 7, 0, 12, 0, 0, 0, 0, 0, 28, 1
Offset: 1

Views

Author

Gary W. Adamson, Jan 19 2007

Keywords

Examples

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

Crossrefs

Cf. A000203 (sigma, diagonal n=k), A007429 (row sums), A051731.

Formula

T(n,k) = Sum_{j=k..n} A051731(n,j)*A130208(j,k) = A051731(n,k)*A000203(k).

A127571 Triangle T(n,k) = phi(n/k)*sigma(k) if k divides n, else 0.

Original entry on oeis.org

1, 1, 3, 2, 0, 4, 2, 3, 0, 7, 4, 0, 0, 0, 6, 2, 6, 4, 0, 0, 12, 6, 0, 0, 0, 0, 0, 8, 4, 6, 0, 7, 0, 0, 0, 15, 6, 0, 8, 0, 0, 0, 0, 0, 13, 4, 12, 0, 0, 6, 0, 0, 0, 0, 18
Offset: 1

Views

Author

Gary W. Adamson, Jan 19 2007

Keywords

Examples

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

Crossrefs

Cf. A000203, A000010, A038040 (row sums), A054523.

Formula

T(n,k) = sum_{j=k..n} A054523(n,j)* A130208(j,k), product of the two infinite lower triangular matrices.
T(n,1) = A000010(n).
T(n,n) = A000203(n).
Showing 1-5 of 5 results.

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