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 1-6 of 6 results.

A106479 First column in inverse of Euler phi sequence matrix.

Original entry on oeis.org

1, -1, -1, 1, -1, 3, -3, -1, 7, -9, 5, -1, -7, 25, -27, -3, 29, -41, 55, -25, -73, 161, -143, -11, 217, -387, 447, -99, -737, 1377, -1219, 209, 1761, -3999, 4357, -1087, -5311, 11463, -12475, 4723, 12705, -33133, 38839, -15005, -35001, 90485, -112395, 54269, 92435, -262555, 328759, -165839, -244533
Offset: 0

Views

Author

Paul Barry, May 03 2005

Keywords

Links

Crossrefs

Cf. A000010, A106476, A106477, A106478, A106480.

Programs

  • Mathematica
    T[n_, k_] := If[k <= n, EulerPhi[n - k + 1], 0]; With[{max = 52}, Inverse[Table[T[n, k], {n, 0, max}, {k, 0, max}]][[;; , 1]]] (* Amiram Eldar, Nov 16 2024 *)

Formula

G.f.: 1/Sum_{k>=0} phi(k+1)*x^k.

A106476 Sequence array of Euler phi function.

Original entry on oeis.org

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

Views

Author

Paul Barry, May 03 2005

Keywords

Comments

Row sums are A002088(n+1). Diagonal sums are A106477. Riordan array (1/Sum_{j>=0} A106479(j)*x^j, x).

Examples

			Triangle begins:
  1;
  1,1;
  2,1,1;
  2,2,1,1;
  4,2,2,1,1;
  2,4,2,2,1,1;

Crossrefs

Cf. A000010, A002088, A106477, A106478, A106479, A106480.

Programs

  • Mathematica
    T[n_, k_] := If[k <= n, EulerPhi[n - k + 1], 0]; Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 16 2024 *)

Formula

Triangle T(n, k) = if(k <= n, phi(n-k+1), 0).

A106478 Inverse of sequence array for Euler phi function.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 3, -1, 1, -1, -1, 1, -3, 3, -1, 1, -1, -1, 1, -1, -3, 3, -1, 1, -1, -1, 1, 7, -1, -3, 3, -1, 1, -1, -1, 1, -9, 7, -1, -3, 3, -1, 1, -1, -1, 1, 5, -9, 7, -1, -3, 3, -1, 1, -1, -1, 1, -1, 5, -9, 7, -1, -3, 3, -1, 1, -1, -1, 1, -7, -1, 5, -9, 7, -1, -3, 3, -1, 1, -1, -1, 1, 25, -7, -1, 5, -9, 7, -1, -3, 3, -1, 1, -1, -1, 1
Offset: 0

Views

Author

Paul Barry, May 03 2005

Keywords

Comments

Row sums are A106480. Sequence matrix for A106479.

Examples

			Triangle begins:
   1;
  -1,  1;
   1, -1, -1,  1;
  -1,  1, -1, -1,  1;
   3, -1,  1, -1, -1,  1;
  -3,  3, -1,  1, -1, -1,  1;
  -1, -3,  3, -1,  1, -1, -1, 1;
   ...

Crossrefs

Cf. A000010 (phi), A106476, A106477, A106479, A106480.

Programs

  • Mathematica
    T[n_, k_] := If[k <= n, EulerPhi[n - k + 1], 0]; With[{max = 14}, Tinv = Inverse[Table[T[n, k], {n, 0, max - 1}, {k, 0, max - 1}]]; Table[Tinv[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten] (* Amiram Eldar, Nov 16 2024 *)

Formula

Riordan array (1/Sum_{j>=0}, phi(j+1) x^j, x).

A106480 Row sums of inverse of sequence array for Euler phi function.

Original entry on oeis.org

1, 0, -1, 0, -1, 2, -1, -2, 5, -4, 1, 0, -7, 18, -9, -12, 17, -24, 31, 6, -67, 94, -49, -60, 157, -230, 217, 118, -619, 758, -461, -252, 1509, -2490, 1867, 780, -4531, 6932, -5543, -820, 11885, -21248, 17591, 2586, -32415, 58070, -54325, -56
Offset: 0

Views

Author

Paul Barry, May 03 2005

Keywords

Crossrefs

Cf. A000010, A106476, A106477, A106478, A106479.

Programs

  • Mathematica
    T[n_, k_] := If[k <= n, EulerPhi[n - k + 1], 0]; With[{max = 48}, Tinv = Inverse[Table[T[n, k], {n, 0, max - 1}, {k, 0, max - 1}]]; Total[Transpose[Table[Tinv[[n, k]], {n, 1, max}, {k, 1, max}]]]] (* Amiram Eldar, Nov 16 2024 *)

Formula

G.f.: 1/(Sum_{j>=0} phi(j+1)*x^j*(1-x)).

A128521 A128174 * A054525 * A000012.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 0, 1, 2, 1, 1, 1, 3, 3, 2, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 1, 3, 3, 3, 3, 2, 2, 1, 1, 0, -1, 1, 2, 2, 3, 2, 2, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Mar 07 2007

Keywords

Comments

Row sums = A106477: (1, 1, 3, 3, 7, 5, 13, 9, 19, 13, ...). A128522 = A054525 * A128174 * A000012.

Examples

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

Crossrefs

Cf. A128174, A054525, A000012, A106477, A128522.

Formula

A128174 * A054525 * A000012 as infinite lower triangular matrices.

A129560 A054523 * A128174.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 2, 0, 1, 7, 0, 1, 0, 1, 5, 4, 1, 1, 0, 1, 13, 0, 1, 0, 1, 0, 1, 9, 6, 1, 2, 0, 1, 0, 1, 19, 0, 3, 0, 1, 0, 1, 0, 1, 13, 10, 1, 2, 1, 1, 0, 1, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Apr 20 2007

Keywords

Comments

Row sums = A002620: (1, 2, 4, 6, 9, 12, 16, 20, ...). Left column = A106477: (1, 1, 3, 3, 7, 5, 13, 9, 19, ...).

Examples

			First few rows of the triangle:
   1;
   1, 1;
   3, 0, 1;
   3, 2, 0, 1;
   7, 0, 1, 0, 1;
   5, 4, 1, 1, 0, 1;
  13, 0, 1, 0, 1, 0, 1;
  ...

Crossrefs

Cf. A054523, A128174, A106477, A002620.

Formula

A054523 * A128174 as infinite lower triangular matrices.
Showing 1-6 of 6 results.

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