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 1-7 of 7 results.

A143230 Triangle read by rows, A130207 * A000012 * A130207.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 2, 2, 4, 4, 4, 4, 8, 8, 16, 2, 2, 4, 4, 8, 4, 6, 6, 12, 12, 24, 12, 36, 4, 4, 8, 8, 16, 8, 24, 16, 6, 6, 12, 12, 24, 12, 36, 24, 36, 4, 4, 8, 8, 16, 8, 24, 16, 24, 16, 10, 10, 20, 20, 40, 20, 60, 40, 60, 40, 100, 4, 4, 8, 8, 16, 8, 24, 16, 24, 16, 40, 16
Offset: 1

Views

Author

Gary W. Adamson, Jul 31 2008

Keywords

Comments

T(n,k) is the number of pairs (a,b), where 0 <= a < n, 0 <= b < k, gcd(a,n) != 1, and gcd(b,k) != 1. - Joerg Arndt, Jun 26 2011

Examples

			First few rows of the triangle:
  1;
  1,  1;
  2,  2,  4;
  2,  2,  4,  4;
  4,  4,  8,  8, 16;
  2,  2,  4,  4,  8,  4;
  6,  6, 12, 12, 24, 12, 36;
  4,  4,  8,  8, 16,  8, 24, 16;
  6,  6, 12, 12, 24, 12, 36, 24, 36;
  ...
T(7,5) = 24 = phi(7) * phi(5) = 6 * 4.

Links

Crossrefs

Cf. A000010, A130207, A143231 (row sums).

Programs

  • Magma
    A143230:= func< n,k | EulerPhi(n)*EulerPhi(k) >;
[A143230(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 10 2024
  • Maple
    with(numtheory): T := proc(n,k) return phi(n)*phi(k): end: seq(seq(T(n,k),k=1..n),n=1..12); # Nathaniel Johnston, Jun 26 2011
  • Mathematica
    A143230[n_, k_]:= EulerPhi[n]*EulerPhi[k];
Table[A143230[n, k], {n, 12}, {k, n}] // Flatten (* G. C. Greubel, Sep 10 2024 *)
  • SageMath
    def A143230(n,k): return euler_phi(n)*euler_phi(k)
flatten([[A143230(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Sep 10 2024

Formula

Triangle read by rows, A130207 * A000012 * A130207, where A130207 = A000010 * 0^(n-k), 1 <= k <= n.
T(n,k) = phi(n) * phi(k), where phi(n) & phi(k) = Euler's totient function.
T(n, 0) = A000010(n) (left border).
Sum_{k=1..n} T(n, k) = A143231(n) (row sums).

A143267 Triangle read by rows, A130207 * A000012 * A127648.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 2, 4, 6, 8, 4, 8, 12, 16, 20, 2, 4, 6, 8, 10, 12, 6, 12, 18, 24, 30, 36, 42, 4, 8, 12, 16, 20, 24, 28, 32, 6, 12, 18, 24, 30, 36, 42, 48, 54, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Offset: 1

Views

Author

Gary W. Adamson, Aug 03 2008

Keywords

Comments

Left border = phi(n), A000010.
Row sums = A143268, phi(n)*T(n): (1, 3, 12, 20, 60, 42,...)

Examples

			First few rows of the triangle =
1;
1, 2;
2, 4, 6;
2, 4, 6, 8;
4, 8, 12, 16, 20;
2, 4, 6, 8, 10, 12;
6, 12, 18, 24, 30, 36, 42;
4, 8, 12, 16, 20, 24, 28, 32;
...
Row 5 = (4, 8, 12, 16, 20) since the first terms of phi(5) = 4; so we perform (4*1, 4*2, 4*3, 4*4, 4*5).

Crossrefs

Cf. A130207, A127648, A000010.

Formula

Triangle read by rows, A130207 * A000012 * A127648; 1<=k<=n. T(n,k) = phi(n)*k.

A143269 Triangle read by rows, A127648 * A000012 * A130207, 1<=k<=n.

Original entry on oeis.org

1, 2, 2, 3, 3, 6, 4, 4, 8, 8, 5, 5, 10, 10, 20, 6, 6, 12, 12, 24, 12, 7, 7, 14, 14, 28, 14, 42, 8, 8, 16, 16, 32, 16, 48, 32, 9, 9, 18, 18, 36, 18, 54, 36, 54, 10, 10, 20, 20, 40, 20, 60, 40, 60, 40, 11, 11, 22, 22, 44, 22, 66, 44, 66, 44, 110
Offset: 1

Views

Author

Gary W. Adamson, Aug 03 2008

Keywords

Comments

Row sums = A143270: (1, 4, 12, 24, 50, 72, 126, 176,...).

Examples

			First few rows of the triangle =
1;
2, 2;
3, 3, 6;
4, 4, 8, 8;
5, 5, 10, 10, 20;
6, 6, 12, 12, 24, 12;
7, 7, 14, 1428, 14, 42;
...
Row 5 = (5, 5, 10, 10, 20) = (5*1, 5*1, 5*2, 5*2, 5*4); where phi(k) = (1, 1, 2, 2, 4,...).

Crossrefs

Cf. A000010, A143270.

Formula

Triangle read by rows, A127648 * A000012 * A130207. T(n,k) = n*phi(k)

A156836 Triangle read by rows, A156348 * A130207.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 2, 0, 2, 1, 0, 0, 0, 4, 1, 3, 6, 0, 0, 2, 1, 0, 0, 0, 0, 0, 6, 1, 4, 0, 8, 0, 0, 0, 4, 1, 0, 12, 0, 0, 0, 0, 0, 6, 1, 5, 0, 0, 20, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 6, 20, 20, 0, 12, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12
Offset: 1

Views

Author

Gary W. Adamson & Mats Granvik, Feb 16 2009

Keywords

Comments

Row sums = A156834: (1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11,...).

Examples

			First few rows of the triangle =
1;
1, 1;
1, 0, 2;
1, 2, 0, 2;
1, 0, 0, 0, 4;
1, 3, 6, 0, 0, 2;
1, 0, 0, 0, 0, 0, 6;
1, 4, 0, 8, 0, 0, 0, 4;
1, 0, 12, 0, 0, 0, 0, 0, 6;
1, 5, 0, 0, 20, 0, 0, 0, 0, 4;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
1, 6, 20, 20, 0, 12, 0, 0, 0, 0, 0, 4;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12;
1, 7, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 6;
...

Crossrefs

Cf. A156348, A130207, A156834, A000010

Formula

Triangle read by rows, A156348 * A130207, where A130207 = an infinite lower
triangular matrix with A000010 as the main diagonal and the rest zeros.

A130209 Diagonalized matrix of d(n), A000005, number of divisors of n.

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 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, 4
Offset: 1

Views

Author

Gary W. Adamson, May 16 2007

Keywords

Examples

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

Links

Crossrefs

Cf. A000005, A130207, A130209.

Programs

  • Maple
    A130209 := proc(n,k)
    if k = n then
        numtheory[tau](n);
    else
        0;
    end if;
end proc: # R. J. Mathar, Aug 06 2016
  • PARI
    A130209(n) = if(ispolygonal(n,3), numdiv((sqrtint(1+(n*8))-1)/2), 0); \\ Antti Karttunen, Jan 17 2025

Formula

T(n,n) = A000005(n),
T(n,k) = 0 if n <> k.

Extensions

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

Views

Author

Gary W. Adamson, May 16 2007

Keywords

Comments

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

Examples

			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;
...

Links

Crossrefs

Cf. A000203, A130207, A130209.

Programs

Formula

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

Extensions

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

A127505 Triangle T(n,k) = mobius(n/k)*phi(k) if k|n, otherwise T(n,k)=0; 1<=k<=n.

Original entry on oeis.org

1, -1, 1, -1, 0, 2, 0, -1, 0, 2, -1, 0, 0, 0, 4, 1, -1, -2, 0, 0, 2, -1, 0, 0, 0, 0, 0, 6, 0, 0, 0, -2, 0, 0, 0, 4, 0, 0, -2, 0, 0, 0, 0, 0, 6, 1, -1, 0, 0, -4, 0, 0, 0, 0, 4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 1, 0, -2, 0, -2, 0, 0
Offset: 1

Views

Author

Gary W. Adamson, Jan 17 2007

Keywords

Examples

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

Crossrefs

Cf. A051731, A000010 (diagonal n=k), A007431 (row sums), A008683 (column k=1).

Formula

T(n,k) = sum_{j=k..n} A054525(n,j)*A130207(j,k), 1<=k<=n.
Showing 1-7 of 7 results.

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