A320275 -id:A320275 - 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.

A329555 Smallest MM-number of a clutter (connected antichain) of n distinct sets.

Original entry on oeis.org

1, 2, 377, 16211, 761917

Offset: 0

Gus Wiseman, Nov 17 2019

nonn

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: {{}}
     377: {{1,2},{1,3}}
   16211: {{1,2},{1,3},{1,4}}
  761917: {{1,2},{1,3},{1,4},{2,3}}

Spanning cutters of distinct sets are counted by A048143.
MM-numbers of connected weak-antichains are A329559.
MM-numbers of sets of sets are A302494.
The smallest BII-number of a clutter with n edges is A329627.
Not requiring the edges to form an antichain gives A329552.
Connected numbers are A305078.
Stable numbers are A316476.
Cf. A056239, A112798, A302242, A319837, A320275, A322113, A327076, A328514, A329552, A329558, A329560, A329561.
Other 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}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
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[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&&stableQ[primeMS[#],Divisible]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

A319837 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 35, 37, 45, 49, 53, 61, 64, 69, 75, 81, 89, 91, 95, 113, 128, 131, 135, 141, 143, 145, 151, 161, 165, 169, 175, 207, 223, 225, 243, 245, 247, 251, 256, 265, 281, 299, 309, 311, 329, 343, 355, 359, 361, 375, 377, 385

Offset: 1

Gus Wiseman, Dec 16 2018

nonn

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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of not necessarily strict antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  19: {{1,1,1}}
  27: {{1},{1},{1}}
  32: {{},{},{},{},{}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  45: {{1},{1},{2}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  61: {{1,2,2}}
  64: {{},{},{},{},{},{}}
  69: {{1},{2,2}}
  75: {{1},{2},{2}}
  81: {{1},{1},{1},{1}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  95: {{2},{1,1,1}}

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A318401, A319721, A320275, A320456.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible]]&]

A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 7, 13, 19, 37, 53, 61, 89, 91, 113, 131, 151, 223, 247, 251, 281, 299, 311, 359, 377, 427, 463, 503, 593, 611, 659, 689, 703, 719, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1291, 1321, 1339, 1363, 1511, 1619, 1703, 1733, 1739, 1757, 1769

Offset: 1

Gus Wiseman, Dec 16 2018

nonn

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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of connected strict antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
    2: {{}}
    3: {{1}}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}

Cf. A003963, A006126, A055932, A056239, A112798, A285573, A286520, A293994, A302242, A318401, A319719, A319837, A320275, A320456, A320532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
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[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==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[#]]&]

A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 7, 13, 15, 19, 35, 37, 53, 61, 69, 89, 91, 95, 113, 131, 141, 143, 145, 151, 161, 165, 223, 247, 251, 265, 281, 299, 309, 311, 329, 355, 359, 377, 385, 407, 427, 437, 463, 503, 591, 593, 611, 655, 659, 667, 671, 689, 703, 719, 721, 759, 791, 827, 851

Offset: 1

Gus Wiseman, Dec 16 2018

nonn

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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of strict antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   2: {{}}
   3: {{1}}
   7: {{1,1}}
  13: {{1,2}}
  15: {{1},{2}}
  19: {{1,1,1}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  53: {{1,1,1,1}}
  61: {{1,2,2}}
  69: {{1},{2,2}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  95: {{2},{1,1,1}}

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A319496, A319721, A319837, A320275, A320456.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible]]&]

cp's OEIS Frontend

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

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A329555 Smallest MM-number of a clutter (connected antichain) of n distinct sets.

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A319837 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers.

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A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

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

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A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

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