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.

Previous Showing 11-18 of 18 results.

A339324 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^9.

Original entry on oeis.org

1, 9, 9, 54, 9, 90, 9, 255, 54, 90, 9, 576, 9, 90, 90, 1035, 9, 576, 9, 576, 90, 90, 9, 2871, 54, 90, 255, 576, 9, 981, 9, 3753, 90, 90, 90, 3861, 9, 90, 90, 2871, 9, 981, 9, 576, 576, 90, 9, 12186, 54, 576, 90, 576, 9, 2871, 90, 2871, 90, 90, 9, 6651, 9, 90, 576
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Comments

Number of factorizations of n into factors (greater than 1) of 9 kinds.

Crossrefs

Cf. A001055, A023008, A301830, A339318, A339319, A339320, A339321, A339322, A339323.

Formula

a(p^k) = A023008(k) for prime p.

A339701 Dirichlet g.f.: Product_{k>=2} (1 - k^(-s))^2.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Dec 13 2020

Keywords

Crossrefs

Cf. A002107, A114592, A301830, A339702, A339703, A339704, A339705, A339706, A339707.

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A301830(n/d) * a(d).
a(n) = Sum_{d|n} A114592(n/d) * A114592(d).
a(p^k) = A002107(k) for prime p.

A301829 Number of ways to choose a nonempty submultiset of a factorization of n into factors greater than one.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 15, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 29, 4, 4, 4, 37, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 64, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 54, 4, 17, 1, 12, 4, 17, 1, 92, 1, 4, 12, 12, 4, 17
Offset: 1

Views

Author

Gus Wiseman, Mar 27 2018

Keywords

Examples

			The a(12) = 12 submultisets ("<" means subset or equal):
(2)<(2*2*3), (3)<(2*2*3), (2*2)<(2*2*3), (2*3)<(2*2*3), (2*2*3)<(2*2*3),
(2)<(2*6), (6)<(2*6), (2*6)<(2*6),
(3)<(3*4), (4)<(3*4), (3*4)<(3*4),
(12)<(12).

Crossrefs

Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301830.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[facs[d]]*Length[facs[n/d]],{d,Rest[Divisors[n]]}],{n,100}]

Formula

a(n) = Sum_{d|n, d>1} f(d) * f(n/d) where f(n) = A001055(n) is the number of factorizations of n into factors greater than 1.

A304796 Number of special sums of integer partitions of n.

Original entry on oeis.org

1, 2, 5, 10, 18, 32, 51, 82, 122, 188, 262, 392, 529, 750, 997, 1404, 1784, 2452, 3123, 4164, 5239, 6916, 8499, 11112, 13693, 17482, 21257, 27162, 32581, 41114, 49606, 61418, 73474, 91086, 107780, 132874, 157359, 191026, 225159, 274110, 320691, 386722, 453875
Offset: 0

Views

Author

Gus Wiseman, May 18 2018

Keywords

Comments

A special sum of an integer partition y is a number n >= 0 such that exactly one submultiset of y sums to n.

Examples

			The a(4) = 18 special positive subset-sums:
0<=(4), 4<=(4),
0<=(22), 2<=(22), 4<=(22),
0<=(31), 1<=(31), 3<=(31), 4<=(31),
0<=(211), 1<=(211), 3<=(211), 4<=(211),
0<=(1111), 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111).

Crossrefs

Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854.

Programs

  • Mathematica
    uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Total],Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]

Formula

a(n) = A301854(n) + A000041(n).

Extensions

More terms from Alois P. Heinz, May 18 2018
a(36)-a(42) from Chai Wah Wu, Sep 26 2023

A349921 Dirichlet g.f.: Product_{k>=2} 1 / (1 - 2 * k^(-s)).

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 2, 14, 6, 6, 2, 18, 2, 6, 6, 34, 2, 18, 2, 18, 6, 6, 2, 46, 6, 6, 14, 18, 2, 22, 2, 74, 6, 6, 6, 58, 2, 6, 6, 46, 2, 22, 2, 18, 18, 6, 2, 114, 6, 18, 6, 18, 2, 46, 6, 46, 6, 6, 2, 70, 2, 6, 18, 166, 6, 22, 2, 18, 6, 22, 2, 150, 2, 6, 18
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 05 2021

Keywords

Crossrefs

Cf. A001055, A061142, A070933, A301830, A349922.

A328705 Dirichlet g.f.: Product_{k>=1} zeta(k*s)^2.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 10, 5, 4, 2, 10, 2, 4, 4, 20, 2, 10, 2, 10, 4, 4, 2, 20, 5, 4, 10, 10, 2, 8, 2, 36, 4, 4, 4, 25, 2, 4, 4, 20, 2, 8, 2, 10, 10, 4, 2, 40, 5, 10, 4, 10, 2, 20, 4, 20, 4, 4, 2, 20, 2, 4, 10, 65, 4, 8, 2, 10, 4, 8, 2, 50, 2, 4, 10, 10, 4, 8, 2, 40
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Comments

Dirichlet convolution of A000688 with itself.

Links

Crossrefs

Cf. A000688, A000712, A001620, A063966, A129667, A188581, A188585, A301830.

Programs

  • Mathematica
    Table[DivisorSum[n, FiniteAbelianGroupCount[n/#] FiniteAbelianGroupCount[#] &], {n, 1, 80}]

Formula

a(n) = Sum_{d|n} A000688(n/d) * A000688(d).
Sum_{k=1..n} a(k) ~ c^2 * n * (log(n) + 2*gamma - 1 - 2*s), where c = A021002 = Product_{k>=2} zeta(k) = 2.2948565916733137941835158313443112887131637994..., s = Sum_{k>=2} k*zeta'(k)/zeta(k) = -2.1955691982567064617939038695473479681910375... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 26 2019
Multiplicative with a(p^e) = A000712(e). - Amiram Eldar, Nov 30 2020

A299764 Number of special products of factorizations of n into factors > 1.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 18, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 32, 6, 6, 6, 44, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 72, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 51, 6, 22, 2, 16, 6, 22, 2, 104, 2, 6, 16, 16, 6
Offset: 1

Views

Author

Gus Wiseman, Jun 08 2018

Keywords

Comments

A special product of a factorization f is a number n > 0 such that exactly one submultiset of f has product n.

Examples

			The a(12) = 16 special subset-products:
1<=(12), 12<=(12),
1<=(2*6), 2<=(2*6), 6<=(2*6), 12<=(2*6),
1<=(3*4), 3<=(3*4), 4<=(3*4), 12<=(3*4),
1<=(2*2*3), 2<=(2*2*3), 3<=(2*2*3), 4<=(2*2*3), 6<=(2*2*3), 12<=(2*2*3).
The a(16) = 18 special subset-products:
1<=(16), 16<=(16),
1<=(4*4), 4<=(4*4), 16<=(4*4),
1<=(2*8), 2<=(2*8), 8<=(2*8), 16<=(2*8),
1<=(2*2*4), 2<=(2*2*4), 8<=(2*2*4), 16<=(2*2*4),
1<=(2*2*2*2), 2<=(2*2*2*2), 4<=(2*2*2*2), 8<=(2*2*2*2), 16<=(2*2*2*2).

Crossrefs

Cf. A001055, A292886, A299701, A301829, A301830, A301854, A301957, A301979, A304796.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sppr[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Times@@#&],Length[#]===1&];
Table[Length[Join@@sppr/@facs[n]],{n,30}]

A329365 Number of factorizations of n into factors (greater than 1) of n kinds.

Original entry on oeis.org

1, 2, 3, 14, 5, 42, 7, 192, 54, 110, 11, 1236, 13, 210, 240, 6460, 17, 3744, 19, 5020, 462, 506, 23, 85176, 350, 702, 4410, 12964, 29, 29730, 31, 604352, 1122, 1190, 1260, 542754, 37, 1482, 1560, 560840, 41, 79422, 43, 47476, 50670, 2162, 47, 15988848, 1274, 68800
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 12 2019

Keywords

Crossrefs

Cf. A001055, A050367, A301830.

Formula

a(p) = p, where p is prime.
Previous Showing 11-18 of 18 results.

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