A329625 -id:A329625 - cp's OEIS frontend

A329552 Smallest MM-number of a connected set of n sets.

1, 2, 39, 195, 5655, 62205, 2674815

Offset: 0

Gus Wiseman, Nov 17 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
        1: {}
        2: {{}}
       39: {{1},{1,2}}
      195: {{1},{2},{1,2}}
     5655: {{1},{2},{1,2},{1,3}}
    62205: {{1},{2},{3},{1,2},{1,3}}
  2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}

MM-numbers of connected set-systems are A328514.
The weight of the system with MM-number n is A302242(n).
Connected numbers are A305078.
Maximum connected divisor is A327076.
BII-numbers of connected sets of sets are A326749.
The smallest BII-number of a connected set of n sets is A329625(n).
Allowing edges to have repeated vertices gives A329553.
Requiring the edges to form an antichain gives A329555.
The smallest MM-number of a set of n nonempty sets is A329557(n).
Cf. A048143, A056239, A112798, A302494, A304714, A304716, A305079, A322389, A328513, A329554, A329556, A329558.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
da=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[da[[Position[PrimeOmega/@da,n][[1,1]]]],{n,First[Split[Union[PrimeOmega/@da],#2==#1+1&]]}]

A329626 Smallest BII-number of an antichain with n edges.

Original entry on oeis.org

0, 1, 3, 11, 139, 820, 2868, 35636, 199476, 723764

Offset: 0

Gus Wiseman, Nov 28 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

A set-system is an antichain if no edge is a proper subset of any other.

			The sequence of terms together with their corresponding set-systems begins:
       0: {}
       1: {{1}}
       3: {{1},{2}}
      11: {{1},{2},{3}}
     139: {{1},{2},{3},{4}}
     820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
    2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
   35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
  199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
  723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}

The connected case is A329627.
The intersecting case is A329628.
BII-numbers of antichains are A326704.
Antichain covers are A006126.
Cf. A048143, A048793, A070939, A303362, A319837, A326031, A326750, A329555, A329560, A329561, A329625.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_]:=!Apply[Or,Outer[#1=!=#2&&SubsetQ[#1,#2]&,u,u,1],{0,1}];
First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&],Length[bpe[#]]&]

A329553 Smallest MM-number of a connected set of n multisets.

Original entry on oeis.org

1, 2, 21, 195, 1365, 25935, 435435

Offset: 0

Gus Wiseman, Nov 17 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
       1: {}
       2: {{}}
      21: {{1},{1,1}}
     195: {{1},{2},{1,2}}
    1365: {{1},{2},{1,1},{1,2}}
   25935: {{1},{2},{1,1},{1,2},{1,1,1}}
  435435: {{1},{2},{1,1},{3},{1,2},{1,3}}

MM-numbers of connected sets of sets are A328514.
The weight of the system with MM-number n is A302242(n).
Connected numbers are A305078.
Maximum connected divisor is A327076.
BII-numbers of connected set-systems are A326749.
The smallest BII-number of a connected set-system is A329625.
The case of strict edges is A329552.
The smallest MM-number of a set of n nonempty sets is A329557(n).
Cf. A056239, A112798, A302494, A305079, A329554, A329555, A329556, A329558.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

A329627 Smallest BII-number of a clutter (connected antichain) with n edges.

Original entry on oeis.org

0, 1, 20, 52, 308, 820, 2868, 68404, 199476, 723764

Offset: 0

Gus Wiseman, Nov 28 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
A set-system is an antichain if no edge is a proper subset of any other.
For n > 1, a(n) appears to be the number whose binary indices are the first n terms of A018900.

			The sequence of terms together with their corresponding set-systems begins:
       0: {}
       1: {{1}}
      20: {{1,2},{1,3}}
      52: {{1,2},{1,3},{2,3}}
     308: {{1,2},{1,3},{2,3},{1,4}}
     820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
    2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
   68404: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5}}
  199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
  723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}

The version for MM-numbers is A329555.
BII-numbers of clutters are A326750.
Clutters of sets are counted by A048143.
Minimum BII-numbers of connected set-systems are A329625.
Minimum BII-numbers of antichains are A329626.
MM-numbers of connected weak antichains of multisets are A329559.
Cf. A048793, A070939, A072639, A320275, A322113, A326031, A326704, A326753, A329628, A329632.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&&Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]

cp's OEIS Frontend

A329552 Smallest MM-number of a connected set of n sets.

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A329626 Smallest BII-number of an antichain with n edges.

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A329553 Smallest MM-number of a connected set of n multisets.

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A329627 Smallest BII-number of a clutter (connected antichain) with n edges.

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