A329553 -id:A329553 - cp's OEIS frontend

A329559 MM-numbers of multiset clutters (connected weak antichains of multisets).

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227

Offset: 1

Gus Wiseman, Nov 18 2019

nonn

A weak antichain of multisets is a multiset of multisets, none of which is a proper subset of any other.

			The sequence of terms tother with their corresponding clutters begins:
   1: {}              37: {{1,1,2}}            91: {{1,1},{1,2}}
   2: {{}}            41: {{6}}                97: {{3,3}}
   3: {{1}}           43: {{1,4}}             101: {{1,6}}
   5: {{2}}           47: {{2,3}}             103: {{2,2,2}}
   7: {{1,1}}         49: {{1,1},{1,1}}       107: {{1,1,4}}
   9: {{1},{1}}       53: {{1,1,1,1}}         109: {{10}}
  11: {{3}}           59: {{7}}               113: {{1,2,3}}
  13: {{1,2}}         61: {{1,2,2}}           121: {{3},{3}}
  17: {{4}}           67: {{8}}               125: {{2},{2},{2}}
  19: {{1,1,1}}       71: {{1,1,3}}           127: {{11}}
  23: {{2,2}}         73: {{2,4}}             131: {{1,1,1,1,1}}
  25: {{2},{2}}       79: {{1,5}}             137: {{2,5}}
  27: {{1},{1},{1}}   81: {{1},{1},{1},{1}}   139: {{1,7}}
  29: {{1,3}}         83: {{9}}               149: {{3,4}}
  31: {{5}}           89: {{1,1,1,2}}         151: {{1,1,2,2}}

Connected numbers are A305078.
Stable numbers are A316476.
Clutters (of sets) are A048143.
Cf. A056239, A112798, A289509, A302242, A302494, A304716, A318991, A319837, A320275, A320456, A328514, A329553, A329555.

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]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],And[stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]<=1]&]

Equals {1} followed by the intersection of A305078 and A316476.

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

Original entry on oeis.org

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&]]}]

A329625 Smallest BII-number of a connected set-system with n edges.

Original entry on oeis.org

0, 1, 5, 7, 23, 31, 63, 127, 383, 511, 1023, 2047, 4095, 8191

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.

			The sequence of terms together with their corresponding set-systems begins:
     0: {}
     1: {{1}}
     5: {{1},{1,2}}
     7: {{1},{2},{1,2}}
    23: {{1},{2},{1,2},{1,3}}
    31: {{1},{2},{1,2},{3},{1,3}}
    63: {{1},{2},{1,2},{3},{1,3},{2,3}}
   127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
   383: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{1,4}}
   511: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4}}

The smallest BII-number of a set-system with n edges is A000225(n).
The smallest BII-number of a set-system with n vertices is A072639(n).
BII-numbers of connected set-systems are A326749.
MM-numbers of connected set-systems are A328514.
The case of clutters is A329627.
Cf. A048143, A048793, A070939, A304716, A305078, A326031, A326753, A329552, A329553, A329626.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],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[0,1000],Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]

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A329559 MM-numbers of multiset clutters (connected weak antichains of multisets).

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

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A329625 Smallest BII-number of a connected set-system with n edges.

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