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.

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A366444 Triangle read by rows: T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

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

Views

Author

Mats Granvik, Oct 12 2023

Keywords

Comments

Sum_{k=1..n} T(n,k) = A063524(n).

Examples

			{
{1}, = 1
{1, -1}, = 0
{2, 0, -2}, = 0
{2, -1, 0, -1}, = 0
{4, 0, 0, 0, -4}, = 0
{2, -2, -2, 0, 0, 2}, = 0
{6, 0, 0, 0, 0, 0, -6}, = 0
{4, -2, 0, -1, 0, 0, 0, -1}, = 0
{6, 0, -4, 0, 0, 0, 0, 0, -2}, = 0
{4, -4, 0, 0, -4, 0, 0, 0, 0, 4}, = 0
{10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10}, = 0
{4, -2, -4, -2, 0, 2, 0, 0, 0, 0, 0, 2} = 0
}

Crossrefs

Cf. A000010, A023900, A366445, A054524, A054525, A063524, A054522, A054523, A129691, A127649.

Programs

  • Mathematica
    nn = 12; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[If[Mod[n, k] == 0, EulerPhi[n/k]*g[k], 0], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

A366445 Triangle read by rows: T(n,k) = A023900(n/k)*phi(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

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

Views

Author

Mats Granvik, Oct 12 2023

Keywords

Comments

Sum_{k=1..n} T(n,k) = A063524(n).

Examples

			{
{1}, = 1
{-1, 1}, = 0
{-2, 0, 2}, = 0
{-1, -1, 0, 2}, = 0
{-4, 0, 0, 0, 4}, = 0
{2, -2, -2, 0, 0, 2}, = 0
{-6, 0, 0, 0, 0, 0, 6}, = 0
{-1, -1, 0, -2, 0, 0, 0, 4}, = 0
{-2, 0, -4, 0, 0, 0, 0, 0, 6}, = 0
{4, -4, 0, 0, -4, 0, 0, 0, 0, 4}, = 0
{-10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 0
{2, 2, -2, -4, 0, -2, 0, 0, 0, 0, 0, 4} = 0
}

Crossrefs

Cf. A000010, A023900, A366444, A054524, A054525, A063524, A054522, A054523, A129691, A127649.

Programs

  • Mathematica
    nn = 12; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[If[Mod[n, k] == 0, g[n/k]*EulerPhi[k], 0], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = A023900(n/k)*phi(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

A127475 Triangle T(n,k) read by rows: T(n,k) = mu(n)*phi(k) if k|n, else T(n,k)=0.

Original entry on oeis.org

1, -1, -1, -1, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, -4, 1, 1, 2, 0, 0, 2, -1, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 4, 0, 0, 0, 0, 4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12
Offset: 1

Views

Author

Gary W. Adamson, Jan 15 2007

Keywords

Examples

			The triangle starts in row n=1 as:
1;
-1, -1;
-1, 0, -2;
0, 0, 0, 0;
-1, 0, 0, 0, -4;
1, 1, 2, 0, 0, 2;
-1, 0, 0, 0, 0, 0, -6;
...

Crossrefs

Cf. A008683, A054522, A055615 (row sums), A097945, A023900.

Formula

T(n,k) = A054522(n,k)*A008683(n).
T(n,1) = A008683(n) = mu(n).
T(n,n) = A097945(n).

A127476 Triangle T(n,k) = sum_{j=k..n, gcd(n,j)=1, k|j} phi(k).

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Jan 15 2007

Keywords

Examples

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

Crossrefs

Cf. A054521, A054522, A023896 (row sums), A000010.

Formula

T(n,k) = sum_{j=k..n} A054521(n,j) * A054522(j,k), product of the two infinite lower triangular matrices.
T(n,1) = A000010(n).
Previous Showing 21-24 of 24 results.

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