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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A167295 Totally multiplicative sequence with a(p) = 3*(p-2) for prime p.

Original entry on oeis.org

1, 0, 3, 0, 9, 0, 15, 0, 9, 0, 27, 0, 33, 0, 27, 0, 45, 0, 51, 0, 45, 0, 63, 0, 81, 0, 27, 0, 81, 0, 87, 0, 81, 0, 135, 0, 105, 0, 99, 0, 117, 0, 123, 0, 81, 0, 135, 0, 225, 0, 135, 0, 153, 0, 243, 0, 153, 0, 171, 0, 177, 0, 135, 0, 297, 0, 195, 0, 189, 0, 207
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165824, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*3^PrimeOmega[n], {n, 1, 100}](* G. C. Greubel, Jun 05 2016 *)
f[p_, e_] := (3*(p-2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Formula

Multiplicative with a(p^e) = (3*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A165824(n) * A166586(n) = 3^bigomega(n) * A166586(n) = 3^A001222(n) * A166586(n).

A167304 Totally multiplicative sequence with a(p) = 3*(p+2) for prime p.

Original entry on oeis.org

1, 12, 15, 144, 21, 180, 27, 1728, 225, 252, 39, 2160, 45, 324, 315, 20736, 57, 2700, 63, 3024, 405, 468, 75, 25920, 441, 540, 3375, 3888, 93, 3780, 99, 248832, 585, 684, 567, 32400, 117, 756, 675, 36288, 129, 4860, 135, 5616, 4725, 900, 147, 311040, 729, 5292
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165824, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*3^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (3*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Formula

Multiplicative with a(p^e) = (3*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)+2))^e(k).
a(n) = A165824(n) * A166590(n) = 3^bigomega(n) * A166590(n) = 3^A001222(n) * A166590(n).

A167313 Totally multiplicative sequence with a(p) = 3*(p-3) for prime p.

Original entry on oeis.org

1, -3, 0, 9, 6, 0, 12, -27, 0, -18, 24, 0, 30, -36, 0, 81, 42, 0, 48, 54, 0, -72, 60, 0, 36, -90, 0, 108, 78, 0, 84, -243, 0, -126, 72, 0, 102, -144, 0, -162, 114, 0, 120, 216, 0, -180, 132, 0, 144, -108, 0, 270, 150, 0, 144, -324, 0, -234, 168, 0, 174, -252, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165824, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*3^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 08 2016 *)
f[p_, e_] := (3*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Formula

Multiplicative with a(p^e) = (3*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165824(n) * A166589(n) = 3^bigomega(n) * A166589(n) = 3^A001222(n) * A166589(n).

A167322 Totally multiplicative sequence with a(p) = 3*(p+3) for prime p.

Original entry on oeis.org

1, 15, 18, 225, 24, 270, 30, 3375, 324, 360, 42, 4050, 48, 450, 432, 50625, 60, 4860, 66, 5400, 540, 630, 78, 60750, 576, 720, 5832, 6750, 96, 6480, 102, 759375, 756, 900, 720, 72900, 120, 990, 864, 81000, 132, 8100, 138, 9450, 7776, 1170, 150, 911250, 900
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165824, A166591.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*3^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (3*(p+3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

Formula

Multiplicative with a(p^e) = (3*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)+3))^e(k).
a(n) = A165824(n) * A166591(n) = 3^bigomega(n) * A166591(n) = 3^A001222(n) * A166591(n).

A354273 Square array read by ascending antidiagonals: A(n,k) = k^Omega(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 4, 1, 1, 2, 9, 4, 5, 1, 1, 4, 3, 16, 5, 6, 1, 1, 2, 9, 4, 25, 6, 7, 1, 1, 8, 3, 16, 5, 36, 7, 8, 1, 1, 4, 27, 4, 25, 6, 49, 8, 9, 1, 1, 4, 9, 64, 5, 36, 7, 64, 9, 10, 1, 1, 2, 9, 16, 125, 6, 49, 8, 81, 10, 11, 1, 1, 8, 3, 16, 25, 216, 7, 64, 9, 100, 11, 12, 1
Offset: 1

Views

Author

Stefano Spezia, May 22 2022

Keywords

Examples

			Array begins:
    1, 1,  1,  1,   1,   1,   1,   1, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 4,  9, 16,  25,  36,  49,  64, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 4,  9, 16,  25,  36,  49,  64, ...
    1, 2,  3,  4,   5,   6,   7,   8, ...
    1, 8, 27, 64, 125, 216, 343, 512, ...
    ...

Links

Crossrefs

Cf. A001222, A051129.
Cf. A000012 (n = 1 or k = 1), A061142 (k = 2), A165824 - A165871 (k = 3..50), A176029 (diagonal).

Programs

  • Mathematica
    A[n_,k_]:=k^PrimeOmega[n]; Flatten[Table[A[n-k+1,k],{n,13},{k,n}]]

Formula

A(n, k) = A051129(A001222(n), k).
The columns are totally multiplicative: A(i*j, k) = A(i, k)*A(j, k).
Previous Showing 11-15 of 15 results.

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