A198085 -id:A198085 - cp's OEIS frontend

A285573 Number of finite nonempty sets of pairwise indivisible divisors of n.

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 9, 2, 5, 5, 5, 2, 9, 2, 9, 5, 5, 2, 14, 3, 5, 4, 9, 2, 19, 2, 6, 5, 5, 5, 19, 2, 5, 5, 14, 2, 19, 2, 9, 9, 5, 2, 20, 3, 9, 5, 9, 2, 14, 5, 14, 5, 5, 2, 49, 2, 5, 9, 7, 5, 19, 2, 9, 5, 19, 2, 34, 2, 5, 9, 9, 5, 19, 2, 20, 5, 5, 2, 49, 5, 5, 5, 14, 2, 49, 5, 9, 5, 5, 5, 27, 2, 9, 9, 19

Offset: 1

Gus Wiseman, Apr 21 2017

nonn

From Robert Israel, Apr 21 2017: (Start)
If n = p^k for prime p, a(n) = k+1.
If n = p^j*q^k for distinct primes p,q, a(n) = binomial(j+k+2,j+1)-1. (End)

			The a(12)=9 sets are: {1}, {2}, {3}, {4}, {6}, {12}, {2,3}, {3,4}, {4,6}.

Cf. A006126, A048143, A076078, A076413, A198085, A285572.

g:= proc(S) local x, Sx; option remember;
   if nops(S) = 0 then return {{}} fi;
   x:= S[1];
   Sx:= subsop(1=NULL,S);
   procname(Sx) union map(t -> t union {x}, procname(remove(s -> s mod x = 0 or x mod s = 0, Sx)))
end proc:
f:= proc(n) local F,D;
  F:= ifactors(n)[2];
  D:= numtheory:-divisors(mul(ithprime(i)^F[i,2],i=1..nops(F)));
  nops(g(D)) - 1;
end proc:
map(f, [$1..100]); # Robert Israel, Apr 21 2017

nn=50;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Rest[stableSets[Divisors[n],Divisible]]],{n,1,nn}]

A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 40, 1344, 203136, 495598592, 309065330371840, 14369391920653644779049472

Offset: 0

Gus Wiseman, May 23 2018

nonn

Only the non-singleton edges are required to form an antichain.

			The a(2) = 4 clutters:
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Cf. A000372, A006126, A014466, A304997, A304999, A306505, A317674, A319721.

For n > 1, a(n) = A048143(n) * 2^n.

A304982 Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.

Original entry on oeis.org

1, 2, 5, 19, 137, 3053, 822526

Offset: 0

Gus Wiseman, May 23 2018

The initial terms 1, 2, 5, 19 are the same as A304981 but the remaining terms differ.

			Non-isomorphic representatives of the a(3) = 19 clutters:
{}
{{1}}
{{1,2}}
{{1,2,3}}
{{2},{1,2}}
{{1,3},{2,3}}
{{3},{1,2,3}}
{{1},{2},{1,2}}
{{1,2},{1,3},{2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{2},{3},{1,2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{3},{1,2},{1,3},{2,3}}
{{2},{3},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Partial sums of A304983.

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 3, 14, 118, 2916, 819473

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 14 clutters:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986, A304997.

Inverse Euler transform of A304997. - Andrew Howroyd, Aug 14 2019

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304986 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n}, if clutters of the form {{x}} are allowed for any vertex x.

Original entry on oeis.org

1, 2, 4, 12, 115, 6834, 7783198, 2414627236078, 56130437209370100252471

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(3) = 12 clutters:
  {}
  {{1}}
  {{2}}
  {{3}}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

a(n > 0) = A198085(n) + 1.

a(n) = A305005(n) + n.

A304981 Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.

Original entry on oeis.org

1, 1, 2, 5, 19, 176, 16118, 489996568

Offset: 0

Gus Wiseman, May 23 2018

nonn

			Non-isomorphic representatives of the a(3) = 5 clutters:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}
Non-isomorphic representatives of the a(4) = 19 clutters:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,2,3,4}}
  {{1,3},{2,3}}
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Partial sums of A261006(n > 0).

A304984 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.

Original entry on oeis.org

1, 2, 7, 56, 1533, 210302, 496838435, 309068803876372, 14369391923126181310256825

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(2) = 7 clutters:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Binomial transform of A304985(n > 0).

A305005 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} without singleton edges.

Original entry on oeis.org

1, 1, 2, 9, 111, 6829, 7783192, 2414627236071, 56130437209370100252463

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(3) = 9 clutters:
  {}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Binomial transform of A048143 if we assume A048143(1) = 0.

a(n) = A198085(n) - n + 1. - Gus Wiseman, Jun 11 2018

A359110 Number of Boolean monoids of order 2^n up to isomorphism.

Original entry on oeis.org

1, 5, 83, 242547

Offset: 1

Choiwah Chow, Dec 18 2022

MathStructures, Boolean Monoids.

Cf. A090802, A066534, A198085, A342908, A058129, A058129.

cp's OEIS Frontend

A285573 Number of finite nonempty sets of pairwise indivisible divisors of n.

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A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

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A304982 Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.

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A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

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A304986 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n}, if clutters of the form {{x}} are allowed for any vertex x.

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A304981 Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.

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A304984 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.

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A305005 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} without singleton edges.

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A359110 Number of Boolean monoids of order 2^n up to isomorphism.

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