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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A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 4, 22, 302, 28630, 216533404, 5592325966377736
Offset: 0

Views

Author

Gus Wiseman, Jul 30 2019

Keywords

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 22 set-systems:
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{}{1}{2}}      {{}{1}{23}}
                 {{}{2}{12}}     {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{1}{2}{3}}
                                 {{}{23}{123}}
                                 {{}{1}{3}{23}}
                                 {{}{2}{3}{123}}
                                 {{}{3}{13}{23}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{1}{2}{3}{23}}
                                 {{}{1}{2}{3}{123}}
                                 {{}{2}{3}{13}{23}}
                                 {{}{1}{3}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{13}{23}}
                                 {{}{1}{2}{3}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}}
                                 {{}{1}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

Crossrefs

The case also closed under union is A001930.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The non-covering case is A108800.
The labeled case is A326881.
Cf. A000798, A102894, A102895, A102897, A193674, A193675, A306445, A326875, A326878, A326880.

Formula

a(n) = A108800(n) - A108800(n-1) for n > 0. - Andrew Howroyd, Aug 10 2019

Extensions

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A339360 Sum of all squarefree numbers with greatest prime factor prime(n).

Original entry on oeis.org

1, 2, 9, 60, 504, 6336, 89856, 1645056, 33094656, 801239040, 24246190080, 777550233600, 29697402470400, 1250501433753600, 55083063155097600, 2649111037319577600, 143390180403000115200, 8619643674791667302400, 534710099148093259776000, 36412881178052121329664000
Offset: 0

Views

Author

Gus Wiseman, Dec 04 2020

Keywords

Examples

			The initial terms are:
   1 = 1,
   2 = 2,
   9 = 3 + 6,
  60 = 5 + 10 + 15 + 30.

Links

Crossrefs

A010036 takes prime indices here to binary indices, row sums of A209862.
A048672 takes prime indices to binary indices in squarefree numbers.
A054640 divides the n-th term by prime(n), row sums of A261144.
A072047 counts prime factors of squarefree numbers.
A339194 is the restriction to semiprimes, row sums of A339116.
A339195 has this as row sums.
A002110 lists primorials.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A056239 is the sum of prime indices of n (Heinz weight).
A246867 groups squarefree numbers by weight, with row sums A147655.
A319246 is the sum of prime indices of the n-th squarefree number.
A319247 lists reversed prime indices of squarefree numbers.
A329631 lists prime indices of squarefree numbers.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014466, A098350, A168472, A282935, A302590, A326878, A338901.

Programs

  • Maple
    f:= proc(n) local i;
  `if`(n=0, 1, ithprime(n)) *mul(1+ithprime(i),i=1..n-1)
end proc:
map(f, [$0..20]); # Robert Israel, Dec 08 2020
  • Mathematica
    Table[Sum[Times@@Prime/@stn,{stn,Select[Subsets[Range[n]],MemberQ[#,n]&]}],{n,10}]

Formula

For n >= 1, a(n) = A054640(n-1) * prime(n).

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 08 2025

A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 6, 32, 418, 23702, 16554476, 1063574497050, 225402367516942398102
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

Examples

			The a(3) = 32 set-systems:
  {}  {{1}}    {{1}{12}}    {{1}{12}{13}}   {{1}{12}{13}{123}}
      {{2}}    {{1}{13}}    {{2}{12}{23}}   {{2}{12}{23}{123}}
      {{3}}    {{2}{12}}    {{3}{13}{23}}   {{3}{13}{23}{123}}
      {{12}}   {{2}{23}}    {{1}{12}{123}}
      {{13}}   {{3}{13}}    {{1}{13}{123}}
      {{23}}   {{3}{23}}    {{2}{12}{123}}
      {{123}}  {{1}{123}}   {{2}{23}{123}}
               {{2}{123}}   {{3}{13}{123}}
               {{3}{123}}   {{3}{23}{123}}
               {{12}{123}}
               {{13}{123}}
               {{23}{123}}

Links

Crossrefs

The case with union instead of intersection is A102896.
The case closed under union and intersection is A326900.
The covering case is A326902.
The connected case is A326903.
The unlabeled version is A326904.
The BII-numbers of these set-systems are A326905.
Cf. A000798, A001930, A006058, A102895, A102898, A182507, A326866, A326876, A326878, A326882.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Formula

a(n) = 1 + Sum_{k=0, n-1} binomial(n,k)*A102895(k). - Andrew Howroyd, Aug 10 2019

Extensions

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A122977 Number of sublattices of the divisor lattice of divisors of n that include n.

Original entry on oeis.org

1, 2, 2, 4, 2, 7, 2, 8, 4, 7, 2, 21, 2, 7, 7, 16, 2, 21, 2, 21, 7, 7, 2, 58, 4, 7, 8, 21, 2, 45, 2, 32, 7, 7, 7, 84, 2, 7, 7, 58, 2, 45, 2, 21, 21, 7, 2, 152, 4, 21, 7, 21, 2, 58, 7, 58, 7, 7, 2, 200, 2, 7, 21, 64, 7, 45, 2, 21, 7, 45, 2, 293, 2, 7, 21, 21, 7, 45, 2, 152, 16, 7, 2, 200, 7, 7
Offset: 1

Views

Author

Franklin T. Adams-Watters, Sep 21 2006

Keywords

Comments

A divisor lattice is closed under GCD and LCM. First differences of A074986. Depends only on the prime signature of n.

Examples

			The a(6) = 7 sublattices of {1,2,3,6} that include 6 are: {6}, {1,6}, {2,6}, {3,6}, {1,2,6}, {1,3,6}, {1,2,3,6}.

Crossrefs

Cf. A051839, A074986, A122978, A122979, A122980, A326878.

Programs

  • Mathematica
    okQ[dd_List] := AllTrue[Subsets[dd, {2}], MemberQ[dd, GCD @@ #] && MemberQ[dd, LCM @@ #]&];
a[n_] := Select[Rest @ Subsets[Divisors[n]], Last[#] == n && okQ[#]&] // Length;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Aug 18 2022 *)

Formula

a(A002110(n)) = A326878(n). - Andrew Howroyd, Apr 17 2020

A326898 Number of unlabeled topologies with up to n points.

Original entry on oeis.org

1, 2, 5, 14, 47, 186, 904, 5439, 41418, 404501, 5122188, 84623842, 1828876351, 51701216248, 1908493827243, 91755916071736, 5729050033597431
Offset: 0

Views

Author

Gus Wiseman, Aug 02 2019

Keywords

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 14 topologies:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{2}{12}     {}{123}
             {}{1}{2}{12}  {}{2}{12}
                           {}{3}{123}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}

Crossrefs

Partial sums of A001930.
The labeled version is A326878.
Cf. A000612, A000798, A003180, A108800, A193675, A306445, A326867, A326882.

A326904 Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 4, 10, 38, 368, 29328, 216591692, 5592326399531792
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.
Apart from the offset the same as A193675. - R. J. Mathar, Aug 09 2019

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{2},{1,2}}  {{1,2,3}}
                          {{2},{1,2}}
                          {{3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{3},{1,3},{2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

Crossrefs

The covering case is A108800(n - 1).
The case with an edge containing all of the vertices is A193674(n - 1).
The case with union instead of intersection is A193674.
The labeled version is A326901.
Cf. A000798, A001930, A006058, A102895, A102898, A326876, A326866, A326878, A326882, A326903, A326906.

Formula

a(n > 0) = 2 * A193674(n - 1).

A326900 Number of set-systems on n vertices that are closed under union and intersection.

Original entry on oeis.org

1, 2, 6, 29, 232, 3032, 62837, 2009408, 97034882, 6952703663, 728107141058, 109978369078580, 23682049666957359, 7195441649260733390, 3056891748255795885338, 1801430622263459795017565, 1462231768717868324127642932, 1624751185398704445629757084188, 2457871026957756859612862822442301
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.

Examples

			The a(0) = 1 through a(3) = 29 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1,2}}      {{3}}
             {{1},{1,2}}  {{1,2}}
             {{2},{1,2}}  {{1,3}}
                          {{2,3}}
                          {{1,2,3}}
                          {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{1,2,3}}
                          {{2},{1,2,3}}
                          {{3},{1,2,3}}
                          {{1,2},{1,2,3}}
                          {{1,3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

Crossrefs

Binomial transform of A006058 (the covering case).
The case closed under union only is A102896.
The case with {} allowed is A306445.
The BII-numbers of these set-systems are A326876.
The case closed under intersection only is A326901.
The unlabeled version is A326908.
Cf. A000798, A001930, A102895, A102898, A326866, A326878, A326882.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}]
(* Second program: *)
A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {, }][[All, 2]];
a[n_] := Sum[Binomial[n, k] A006058[[k + 1]], {k, 0, n}];
a /@ Range[0, 18] (* Jean-François Alcover, Jan 01 2020 *)

Extensions

a(16)-a(18) from A006058 by Jean-François Alcover, Jan 01 2020

A326902 Number of set-systems (without {}) covering n vertices that are closed under intersection.

Original entry on oeis.org

1, 1, 3, 19, 319, 21881, 16417973, 1063459099837, 225402359008808647339
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

Examples

			The a(0) = 1 through a(3) = 19 set-systems:
  {}  {{1}}  {{1,2}}      {{1,2,3}}
             {{1},{1,2}}  {{1},{1,2,3}}
             {{2},{1,2}}  {{2},{1,2,3}}
                          {{3},{1,2,3}}
                          {{1,2},{1,2,3}}
                          {{1,3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{1},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

Crossrefs

The case closed under union and intersection is A006058.
The case with union instead of intersection is A102894.
The unlabeled version is A108800(n - 1).
The non-covering case is A326901.
The connected case is A326903.
Cf. A000798, A001930, A102895, A102898, A182507, A326866, A326876, A326878, A326882, A326900, A326901, A326904.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Formula

Inverse binomial transform of A326901. - Andrew Howroyd, Aug 10 2019

Extensions

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.

Original entry on oeis.org

0, 1, 3, 16, 209, 11851, 8277238, 531787248525, 112701183758471199051
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.
If {} is allowed, we get Moore families (A102896, cf A102895).

Examples

			The a(1) = 1 through a(3) = 16 set-systems:
  {{1}}  {{1,2}}      {{1,2,3}}
         {{1},{1,2}}  {{1},{1,2,3}}
         {{2},{1,2}}  {{2},{1,2,3}}
                      {{3},{1,2,3}}
                      {{1,2},{1,2,3}}
                      {{1,3},{1,2,3}}
                      {{2,3},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1},{1,3},{1,2,3}}
                      {{2},{1,2},{1,2,3}}
                      {{2},{2,3},{1,2,3}}
                      {{3},{1,3},{1,2,3}}
                      {{3},{2,3},{1,2,3}}
                      {{1},{1,2},{1,3},{1,2,3}}
                      {{2},{1,2},{2,3},{1,2,3}}
                      {{3},{1,3},{2,3},{1,2,3}}

Links

Crossrefs

The case closed under union and intersection is A006058.
The case with union instead of intersection is A102894.
The unlabeled version is A193674.
The case without requiring the maximum edge is A326901.
The covering case is A326902.
Cf. A000798, A001930, A102895, A102898, A326866, A326876, A326878, A326882, A326904.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],MemberQ[#,Range[n]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Formula

a(n) = A326901(n) / 2 for n > 0. - Andrew Howroyd, Aug 10 2019

Extensions

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A326908 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and intersection.

Original entry on oeis.org

2, 4, 9, 23, 70, 256, 1160, 6599, 48017, 452518, 5574706, 90198548, 1919074899, 53620291147, 1962114118390, 93718030190126, 5822768063787557
Offset: 0

Views

Author

Gus Wiseman, Aug 03 2019

Keywords

Examples

			Non-isomorphic representatives of the a(0) = 2 through a(3) = 23 sets of subsets:
  {}    {}       {}              {}
  {{}}  {{}}     {{}}            {{}}
        {{1}}    {{1}}           {{1}}
        {{}{1}}  {{12}}          {{12}}
                 {{}{1}}         {{}{1}}
                 {{}{12}}        {{123}}
                 {{2}{12}}       {{}{12}}
                 {{}{2}{12}}     {{}{123}}
                 {{}{1}{2}{12}}  {{2}{12}}
                                 {{3}{123}}
                                 {{}{2}{12}}
                                 {{23}{123}}
                                 {{}{3}{123}}
                                 {{}{23}{123}}
                                 {{}{1}{2}{12}}
                                 {{3}{23}{123}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{3}{13}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

Crossrefs

The labeled version is A306445.
Taking first differences and prepending 1 gives A326898.
Taking second differences and prepending two 1's gives A001930.
Cf. A000612, A000798, A003180, A108798, A108800, A193675, A326867, A326876, A326878, A326882, A326883.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union@@@Tuples[#,2]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
Previous Showing 11-20 of 23 results. Next

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