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 1-7 of 7 results.

A318128 Number of set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 0, 2, 84, 31478, 2147000136, 9223371998203475474, 170141183460469231537996491257596836636, 57896044618658097711785492504343953922551603929769020459976077632195066756398
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			The a(2) = 2 set-systems are {{1},{2}}, and {{1},{2},{1,2}}.

Crossrefs

Cf. A000371, A001146, A003465, A119563, A131288.
Cf. A318128, A318129, A318130, A318131, A318132.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#===Range[n],Intersection@@#=={}]&]],{n,4}]

Formula

Inverse binomial transform of A318129.

A318129 Number of sets of nonempty subsets of {1,...,n} with intersection {}.

Original entry on oeis.org

1, 1, 3, 91, 31827, 2147158387, 9223372011085950171, 170141183460469231602560095290109272523, 57896044618658097711785492504343953923912733397452774312538303978325772978595
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			The a(2) = 3 sets of sets are {}, {{1},{2}}, {{1},{2},{1,2}}.

Crossrefs

Cf. A000371, A001146, A003465,A040996, A119563, A131288.
Cf. A318128, A318130, A318131, A318132.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]],Or[#=={},Intersection@@#=={}]&]],{n,0,4}]

Formula

Binomial transform of A318128.
a(n) = A318130(n) - 2^(2^n - 1). [corrected]

A318131 Number of non-isomorphic sets of finite (possibly empty) sets with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 1, 6, 60, 3836, 37325360, 25626412263611792, 67516342973185974276922865448446208, 2871827610052485009904013737758920847534777143951264797898686184985092096
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			Non-isomorphic representatives of the a(2) = 6 sets of sets:
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Links

Crossrefs

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.
Cf. A318128, A318129, A318130, A318132.

Programs

  • Mathematica
    sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Subsets[Range[n]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

Formula

a(n) = 2*(A055621(n) - A055621(n-1)) = 2*(A000612(n) - 2*A000612(n-1) + A000612(n-2)) for n >= 2. - Andrew Howroyd, Jan 29 2024

Extensions

a(5) onwards from Andrew Howroyd, Jan 29 2024

A318132 Number of non-isomorphic set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 0, 2, 26, 1884, 18660728, 12813206113141264, 33758171486592987125648226573752576, 1435913805026242504952006868879460423733630400489039411798068453617852416
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 26 set-systems:
  {{1},{2,3}}
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Links

Crossrefs

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.
Cf. A318128, A318129, A318130, A318131.

Programs

  • Mathematica
    sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

Formula

a(n) = A055621(n) - 2*A055621(n-1) = A000612(n) - 3*A000612(n-1) + 2*A000612(n-2) for n >= 2. - Andrew Howroyd, Jan 29 2024

Extensions

a(5) onwards from Andrew Howroyd, Jan 29 2024

A318130 Number of sets of subsets of {1,...,n} with intersection {}.

Original entry on oeis.org

2, 3, 11, 219, 64595, 4294642035, 18446744047940725979, 340282366920938463334247399005993378251, 115792089237316195423570985008687907850547725730273056332267095982282337798563
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			The a(2) = 11 sets of sets:
  {}
  {{}}
  {{},{1}}
  {{},{2}}
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Crossrefs

Cf. A000371, A001146, A003465, A119563, A131288, A318128, A318129.
Cf. A318128, A318129, A318131, A318132.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n]]],Or[#=={},Intersection@@#=={}]&]],{n,0,4}]

Formula

Binomial transform of A131288.
Inverse binomial transform of A119563(n) = 2^(2^n) + 2^n - 1.

A119564 Define F(n) = 2^(2^n)+1 = n-th Fermat number, M(n) = 2^n-1 = the n-th Mersenne number. Then a(n) = F(n)-M(n)-1 = 2^(2^n) - 2^n + 1.

Original entry on oeis.org

2, 3, 13, 249, 65521, 4294967265, 18446744073709551553, 340282366920938463463374607431768211329, 115792089237316195423570985008687907853269984665640564039457584007913129639681
Offset: 0

Views

Author

Cino Hilliard, May 31 2006

Keywords

Comments

The numbers n that divide a(n) are A373580. - Thomas Ordowski, Jun 11 2024

Examples

			F(2) = 2^(2^2)+1 = 17, M(2) = 2^2-1 = 3, F(2)-M(2)-1 = 13.

Crossrefs

Cf. A119550, A119563, A307843, A373580.

Programs

  • PARI
    fm4(n) = for(x=0,n,y=2^(2^x)+1-(2^x-1)-1;print1(y","))

Formula

a(n) = (2^(2^n) - 1) - (2^n - 2). - Thomas Ordowski, Jun 11 2024

Extensions

Edited by N. J. A. Sloane, Jun 03 2006
Definition corrected by R. J. Mathar, May 15 2007

A119550 Prime numbers of the form 2^(2^k) + 2^k - 1.

Original entry on oeis.org

2, 5, 19, 263, 65551
Offset: 1

Views

Author

Cino Hilliard, May 31 2006

Keywords

Examples

			F(2)= 2^(2^2)+1 = 17, M(2) = 2^2-1 = 3, F(2)+ M(2)-1 = 19 is prime, so 2 is a member.

Crossrefs

Cf. A119563, A119564.

Programs

  • Mathematica
    Select[Table[2^(2^k)+2^k-1,{k,0,10}],PrimeQ] (* James C. McMahon, Sep 15 2024 *)
  • PARI
    fmp3(n)=for(x=0,n,y=2^(2^x)+2^x-1;if(ispseudoprime(y),print1(y",")))

Formula

Define F(n) = 2^(2^n)+1 = n-th Fermat number, M(n) = 2^n-1 = the n-th Mersenne number. Then we are considering the numbers f(n) = F(n)+M(n)-1 = 2^(2^n) + 2^n - 1 (cf. A119563).

Extensions

Edited by N. J. A. Sloane, Jun 03 2006
Definition corrected by Stefan Steinerberger, Jun 10 2007
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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