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 1-8 of 8 results.

A339318 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^3.

Original entry on oeis.org

1, 3, 3, 9, 3, 12, 3, 22, 9, 12, 3, 39, 3, 12, 12, 51, 3, 39, 3, 39, 12, 12, 3, 105, 9, 12, 22, 39, 3, 57, 3, 108, 12, 12, 12, 135, 3, 12, 12, 105, 3, 57, 3, 39, 39, 12, 3, 258, 9, 39, 12, 39, 3, 105, 12, 105, 12, 12, 3, 201, 3, 12, 39, 221, 12, 57, 3, 39, 12, 57
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 3 kinds.

Examples

			From _Antti Karttunen_, Dec 15 2021: (Start)
For n = 8, A001055(8) = 3, as it has three ordinary factorizations: (8), (4*2), (2*2*2). When allowing each of the factors appear in three different guises (here indicated with a subscript), and where neither the order of factors nor their subscripts matter, we get the following 22 different factorizations:
  (8_3), (8_2), (8_1),
  (4_3 * 2_3), (4_3 * 2_2), (4_3 * 2_1),
  (4_2 * 2_3), (4_2 * 2_2), (4_2 * 2_1),
  (4_1 * 2_3), (4_1 * 2_2), (4_1 * 2_1),
  (2_3 * 2_3 * 2_3), (2_3 * 2_3 * 2_2), (2_3 * 2_3 * 2_1),
  (2_3 * 2_2 * 2_2), (2_3 * 2_2 * 2_1), (2_3 * 2_1 * 2_1),
  (2_2 * 2_2 * 2_2), (2_2 * 2_2 * 2_1), (2_2 * 2_1 * 2_1),
  (2_1 * 2_1 * 2_1),
therefore a(8) = 22. (End)

Links

Crossrefs

Cf. A000716, A001055, A301830, A339319, A339320, A339321, A339322, A339323, A339324.

Programs

Formula

a(p^k) = A000716(k) for prime p.

A339319 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^4.

Original entry on oeis.org

1, 4, 4, 14, 4, 20, 4, 40, 14, 20, 4, 76, 4, 20, 20, 105, 4, 76, 4, 76, 20, 20, 4, 236, 14, 20, 40, 76, 4, 116, 4, 252, 20, 20, 20, 306, 4, 20, 20, 236, 4, 116, 4, 76, 76, 20, 4, 656, 14, 76, 20, 76, 4, 236, 20, 236, 20, 20, 4, 476, 4, 20, 76, 574, 20, 116, 4, 76, 20, 116
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 4 kinds.

Crossrefs

Cf. A001055, A023003, A301830, A339318, A339320, A339321, A339322, A339323, A339324.

Formula

a(p^k) = A023003(k) for prime p.

A339320 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^5.

Original entry on oeis.org

1, 5, 5, 20, 5, 30, 5, 65, 20, 30, 5, 130, 5, 30, 30, 190, 5, 130, 5, 130, 30, 30, 5, 455, 20, 30, 65, 130, 5, 205, 5, 506, 30, 30, 30, 595, 5, 30, 30, 455, 5, 205, 5, 130, 130, 30, 5, 1405, 20, 130, 30, 130, 5, 455, 30, 455, 30, 30, 5, 955, 5, 30, 130, 1265, 30, 205, 5, 130
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 5 kinds.

Crossrefs

Cf. A001055, A023004, A301830, A339318, A339319, A339321, A339322, A339323, A339324.

Formula

a(p^k) = A023004(k) for prime p.

A339321 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^6.

Original entry on oeis.org

1, 6, 6, 27, 6, 42, 6, 98, 27, 42, 6, 204, 6, 42, 42, 315, 6, 204, 6, 204, 42, 42, 6, 792, 27, 42, 98, 204, 6, 330, 6, 918, 42, 42, 42, 1044, 6, 42, 42, 792, 6, 330, 6, 204, 204, 42, 6, 2682, 27, 204, 42, 204, 6, 792, 42, 792, 42, 42, 6, 1716, 6, 42, 204, 2492, 42, 330
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 6 kinds.

Crossrefs

Cf. A001055, A023005, A301830, A339318, A339319, A339320, A339322, A339323, A339324.

Formula

a(p^k) = A023005(k) for prime p.

A339322 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^7.

Original entry on oeis.org

1, 7, 7, 35, 7, 56, 7, 140, 35, 56, 7, 301, 7, 56, 56, 490, 7, 301, 7, 301, 56, 56, 7, 1281, 35, 56, 140, 301, 7, 497, 7, 1547, 56, 56, 56, 1701, 7, 56, 56, 1281, 7, 497, 7, 301, 301, 56, 7, 4711, 35, 301, 56, 301, 7, 1281, 56, 1281, 56, 56, 7, 2849, 7, 56, 301, 4522
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 7 kinds.

Crossrefs

Cf. A001055, A023006, A301830, A339318, A339319, A339320, A339321, A339323, A339324.

Formula

a(p^k) = A023006(k) for prime p.

A339323 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^8.

Original entry on oeis.org

1, 8, 8, 44, 8, 72, 8, 192, 44, 72, 8, 424, 8, 72, 72, 726, 8, 424, 8, 424, 72, 72, 8, 1960, 44, 72, 192, 424, 8, 712, 8, 2464, 72, 72, 72, 2620, 8, 72, 72, 1960, 8, 712, 8, 424, 424, 72, 8, 7768, 44, 424, 72, 424, 8, 1960, 72, 1960, 72, 72, 8, 4456, 8, 72, 424
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 8 kinds.

Crossrefs

Cf. A001055, A023007, A301830, A339318, A339319, A339320, A339321, A339322, A339324.

Formula

a(p^k) = A023007(k) for prime p.

A339341 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^9.

Original entry on oeis.org

1, 9, 9, 45, 9, 90, 9, 174, 45, 90, 9, 495, 9, 90, 90, 576, 9, 495, 9, 495, 90, 90, 9, 2061, 45, 90, 174, 495, 9, 981, 9, 1701, 90, 90, 90, 2961, 9, 90, 90, 2061, 9, 981, 9, 495, 495, 90, 9, 7245, 45, 495, 90, 495, 9, 2061, 90, 2061, 90, 90, 9, 5841, 9, 90, 495
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Crossrefs

Cf. A022574, A045778, A328706, A339324, A339335, A339336, A339337, A339338, A339339, A339340.

Formula

a(p^k) = A022574(k) for prime p.

A339716 Dirichlet g.f.: Product_{k>=2} (1 - k^(-s))^9.

Original entry on oeis.org

1, -9, -9, 27, -9, 72, -9, -12, 27, 72, -9, -171, -9, 72, 72, -90, -9, -171, -9, -171, 72, 72, -9, -63, 27, 72, -12, -171, -9, -495, -9, 135, 72, 72, 72, 189, -9, 72, 72, -63, -9, -495, -9, -171, -171, 72, -9, 747, 27, -171, 72, -171, -9, -63, 72, -63, 72, 72, -9, 801
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 14 2020

Keywords

Crossrefs

Cf. A010817, A114592, A339324, A339701, A339702, A339703, A339704, A339705, A339706, A339707.

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339324(n/d) * a(d).
a(p^k) = A010817(k) for prime p.
Showing 1-8 of 8 results.

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