A329558 -id:A329558 - cp's OEIS frontend

A329557 Smallest MM-number of a set of n nonempty sets.

1, 3, 15, 165, 2145, 36465, 1057485, 32782035, 1344063435, 57794727705, 2716352202135, 160264779925965, 10737740255039655, 783855038617894815, 61924548050813690385, 5139737488217536301955, 519113486309971166497455, 56583370007786857148222595, 6393920810879914857749153235

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: {}
        3: {{1}}
       15: {{1},{2}}
      165: {{1},{2},{3}}
     2145: {{1},{2},{3},{1,2}}
    36465: {{1},{2},{3},{1,2},{4}}
  1057485: {{1},{2},{3},{1,2},{4},{1,3}}

MM-numbers of sets of sets are A302494.
MM-numbers of sets of nonempty sets are A329629.
The version allowing empty sets is A329558.
The version without singletons is A329554.
Cf. A056239, A072639, A112798, A302242, A326031, A329552, A329555, A329556.
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}]]]];
dae=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],1]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

a(n) = my(k=1); prod(i=1, n, until(issquarefree(k), k++); prime(k)); \\ Jinyuan Wang, Feb 23 2025

a(n) = A329558(n + 1)/2.

More terms from Jinyuan Wang, Feb 23 2025

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&]]}]

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&]]}]

A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

Original entry on oeis.org

1, 13, 377, 16211, 761917, 55619941, 4393975339, 443791509239, 50148440544007, 6870336354528959, 954976753279525301, 142291536238649269849, 23193520406899830985387, 3873317907952271774559629, 701070541339361191195292849, 139513037726532877047863276951

Offset: 0

Gus Wiseman, Nov 17 2019

nonn

A multiset multisystem is a finite multiset of finite multisets. 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}}.

			The sequence of terms together with their corresponding systems begins:
       1: {}
      13: {{1,2}}
     377: {{1,2},{1,3}}
   16211: {{1,2},{1,3},{1,4}}
  761917: {{1,2},{1,3},{1,4},{2,3}}

The smallest BII-number of a set of n sets is A000225(n).
BII-numbers of set-systems with no singletons are A326781.
MM-numbers of sets of nonempty sets are the odd terms of A302494.
MM-numbers of multisets of nonempty non-singleton sets are A320629.
The version with empty edges is A329556.
The version with singletons is A329557.
The version with empty edges and singletons is A329558.
Cf. A056239, A072639, A112798, A279952, A302242, A326031, A329552, A329556.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

sqvs=Select[Range[2,30],SquareFreeQ[#]&&!PrimeQ[#]&];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]

a(n) = Product_{i = 1..n} prime(A120944(i)).

A329556 Smallest MM-number of a set of n sets with no singletons.

Original entry on oeis.org

1, 2, 26, 754, 32422, 1523834

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: {{}}
       26: {{},{1,2}}
      754: {{},{1,2},{1,3}}
    32422: {{},{1,2},{1,3},{1,4}}
  1523834: {{},{1,2},{1,3},{1,4},{2,3}}

MM-numbers of sets of sets with no singletons are A329630.
The case without empty edges is A329554.
MM-numbers of sets of sets are A302494.
Cf. A056239, A112798, A302242, A329552, A329557, 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}]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],_?PrimeQ]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

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&]]}]

cp's OEIS Frontend

A329557 Smallest MM-number of a set of n nonempty sets.

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

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

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A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

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A329556 Smallest MM-number of a set of n sets with no singletons.

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

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