A286520 -id:A286520 - cp's OEIS frontend

A303838 Number of z-forests with least common multiple n > 1.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 8, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 8, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 16, 1, 2, 3, 1, 2, 8, 1, 3, 2, 8, 1, 7, 1, 2, 3, 3, 2, 8, 1, 5, 1, 2, 1, 16, 2, 2

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-forest is a finite set of pairwise indivisible positive integers greater than 1 such that all connected components are z-trees, meaning they have clutter density -1.
This is a generalization to multiset systems of the usual definition of hyperforest (viz. hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).
If n is squarefree with k prime factors, then a(n) = A134954(k).
Differs from A324837 at positions {1, 180, 210, ...}. For example, a(210) = 55, A324837(210) = 49.

			The a(60) = 16 z-forests together with the corresponding multiset systems (see A112798, A302242) are the following.
       (60): {{1,1,2,3}}
     (3,20): {{2},{1,1,3}}
     (4,15): {{1,1},{2,3}}
     (4,30): {{1,1},{1,2,3}}
     (5,12): {{3},{1,1,2}}
     (6,20): {{1,2},{1,1,3}}
    (10,12): {{1,3},{1,1,2}}
    (12,15): {{1,1,2},{2,3}}
    (12,20): {{1,1,2},{1,1,3}}
    (15,20): {{2,3},{1,1,3}}
    (3,4,5): {{2},{1,1},{3}}
   (3,4,10): {{2},{1,1},{1,3}}
    (4,5,6): {{1,1},{3},{1,2}}
   (4,6,10): {{1,1},{1,2},{1,3}}
   (4,6,15): {{1,1},{1,2},{2,3}}
  (4,10,15): {{1,1},{1,3},{2,3}}

Gus Wiseman, Table of n, a(n) for n = 1..250
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A304118.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[Rest[Subsets[Rest[Divisors[n]]]],Function[s,LCM@@s==n&&And@@Table[zensity[Select[s,Divisible[m,#]&]]==-1,{m,zsm[s]}]&&Select[Tuples[s,2],UnsameQ@@#&&Divisible@@#&]=={}]]],{n,100}]

A218970 Number of connected cyclic conjugacy classes of subgroups of the symmetric group.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 5, 3, 8, 2, 14, 3, 17, 11, 24, 10, 40, 16, 53, 35, 71, 43, 112, 68, 144, 112, 203, 152, 301, 219, 393, 342, 540, 474, 770, 661, 1022, 967, 1397, 1313, 1928, 1821, 2565, 2564, 3439, 3445, 4676, 4687, 6186, 6406, 8215, 8543, 10974, 11435

Offset: 0

Liam Naughton, Nov 26 2012

nonn

a(n) is also the number of connected partitions of n in the following sense. Given a partition of n, the vertices are the parts of the partition and two vertices are connected if and only if their gcd is greater than 1. We call a partition connected if the graph is connected.

			From _Gus Wiseman_, Dec 03 2018: (Start)
The a(12) = 14 connected integer partitions of 12:
  (12)  (6,6)   (4,4,4)  (3,3,3,3)  (4,2,2,2,2)  (2,2,2,2,2,2)
        (8,4)   (6,3,3)  (4,4,2,2)
        (9,3)   (6,4,2)  (6,2,2,2)
        (10,2)  (8,2,2)
(End)

Liam Naughton and Goetz Pfeiffer, Integer sequences realized by the subgroup pattern of the symmetric group, arXiv:1211.1911 [math.GR], 2012-2013.
Liam Naughton, CountingSubgroups.g
Liam Naughton and Goetz Pfeiffer, Tomlib, The GAP table of marks library

Cf. A018783, A200976, A286518, A286520, A290103, A304714, A304716, A305078, A305079, A322306, A322307.

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]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],Length[zsm[#]]==1&]],{n,10}]

For n > 1, a(n) = A304716(n) - 1. - Gus Wiseman, Dec 03 2018

More terms from Gus Wiseman, Dec 03 2018

A304717 Number of connected strict integer partitions of n with pairwise indivisible parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 2, 4, 3, 5, 2, 5, 4, 6, 3, 7, 6, 9, 5, 9, 8, 13, 10, 15, 9, 15, 13, 18, 14, 22, 21, 26, 19, 29, 24, 36, 31, 40, 35, 45, 38, 54, 55, 59, 55, 70, 69, 84, 74, 89, 86, 107, 103, 119, 115, 143, 143, 159

Offset: 1

Gus Wiseman, May 17 2018

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

			The a(34) = 13 connected strict integer partitions with pairwise indivisible parts are (34), (18,16), (20,14), (22,12), (24,10), (26,8), (28,6), (30,4), (14,12,8), (15,10,9), (20,8,6), (14,10,6,4), (15,9,6,4). Their corresponding multiset multisystems (see A112798, A302242) are the following.
         (34): {{1,7}}
       (30 4): {{1,2,3},{1,1}}
       (28 6): {{1,1,4},{1,2}}
       (26 8): {{1,6},{1,1,1}}
      (24 10): {{1,1,1,2},{1,3}}
      (22 12): {{1,5},{1,1,2}}
      (20 14): {{1,1,3},{1,4}}
     (20 8 6): {{1,1,3},{1,1,1},{1,2}}
      (18 16): {{1,2,2},{1,1,1,1}}
    (15 10 9): {{2,3},{1,3},{2,2}}
   (15 9 6 4): {{2,3},{2,2},{1,2},{1,1}}
    (14 12 8): {{1,4},{1,1,2},{1,1,1}}
  (14 10 6 4): {{1,4},{1,3},{1,2},{1,1}}

Cf. A000009, A003963, A006126, A048143, A051424, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A293993, A302242, A303362, A304714, A304716.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c==={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]===1&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,30}]

A305193 Number of connected factorizations of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 10, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 12, 1, 2, 2, 5, 1, 1, 1, 4, 1

Offset: 1

Gus Wiseman, May 27 2018

nonn

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. This sequence counts factorizations S such that G(S) is a connected graph.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 07 2018

			The a(72) = 10 factorizations:
(72),
(2*2*18), (2*3*12), (2*6*6), (3*4*6),
(2*36), (3*24), (4*18), (6*12),
(2*2*3*6).

Antti Karttunen, Table of n, a(n) for n = 1..20736
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A048143, A281116, A286518, A286520, A290103, A303837, A304118, A304714, A304716, A305078, A305079, A319786.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[zsm[#]]==1&]],{n,100}]

is_connected(facs) = { my(siz=length(facs)); if(1==siz,1,my(m=matrix(siz,siz,i,j,(gcd(facs[i],facs[j])!=1))^siz); for(n=1,siz,if(0==vecmin(m[n,]),return(0))); (1)); };
A305193aux(n, m, facs) = if(1==n, is_connected(Set(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A305193aux(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Nov 07 2018
A305193(n) = if(1==n,0,A305193aux(n, n, List([]))); \\ Antti Karttunen, Nov 07 2018

More terms from Antti Karttunen, Nov 07 2018

A304382 Number of z-trees summing to n. Number of connected strict integer partitions of n with pairwise indivisible parts and clutter density -1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 2, 4, 3, 5, 2, 5, 4, 6, 3, 7, 6, 8, 4, 9, 8, 13, 9, 15, 8, 14, 12, 16, 12, 20, 20, 24, 15, 27, 20, 33, 27, 35

Offset: 1

Gus Wiseman, May 21 2018

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

The clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(LCM(S)).

			The a(30) = 8 z-trees together with the corresponding multiset systems are the following.
       (30): {{1,2,3}}
     (26,4): {{1,6},{1,1}}
     (22,8): {{1,5},{1,1,1}}
     (21,9): {{2,4},{2,2}}
    (16,14): {{1,1,1,1},{1,4}}
   (15,9,6): {{2,3},{2,2},{1,2}}
  (14,10,6): {{1,4},{1,3},{1,2}}
  (12,10,8): {{1,1,2},{1,3},{1,1,1}}

Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A000009, A006126, A048143, A054921, A112798, A285572, A286518, A286520, A293993, A303362, A303837, A304714, A304716, A305194, A305195.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
strConnAnti[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]==1&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}&];
Table[Length[Select[strConnAnti[n],Length[#]==1||zreeQ[#]&]],{n,20}]

A317634 Number of caps (also clutter partitions) of clutters (connected antichains) spanning n vertices.

Original entry on oeis.org

1, 0, 1, 9, 315, 64880

Offset: 0

Gus Wiseman, Aug 02 2018

A kernel of a clutter is the restriction of the edge set to all edges contained within some connected vertex set. A clutter partition is a set partition of the edge set using kernels.

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

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
Gus Wiseman, All clutter partitions of non-isomorphic clutters on 4 vertices.

Cf. A001187, A030019, A048143, A275307, A286520,, A293510, A304717, A317631, A317632, A317635.

A317635 Number of connected vertex sets of clutters (connected antichains) spanning n vertices.

Original entry on oeis.org

1, 0, 1, 14, 486, 71428

Offset: 0

Gus Wiseman, Aug 02 2018

A connected vertex set in a clutter is any union of a connected subset of the edges.

			There are four connected vertex sets of {{1,2},{1,3},{2,3}}, namely {1,2,3}, {1,2}, {1,3}, {2,3}; there are three connected vertex sets of {{1,2},{1,3}}, {{1,2},{2,3}}, and {{1,3},{2,3}} each; and there is one connected vertex set of {{1,2,3}}. So we have a total of a(3) = 4 + 3 * 3 + 1 = 14 connected vertex sets.

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A001187, A006126, A030019, A048143, A134954, A275307, A286520, A293510, A304717, A317631, A317632, A317634.

nn=5;
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
clutQ[eds_]:=And[UnsameQ@@eds,!Apply[Or,Outer[#1=!=#2&&Complement[#1,#2]=={}&,eds,eds,1],{0,1}],Length[csm[eds]]==1];
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]]]];
swell[c_]:=Union@@FixedPointList[Union[ReplaceList[#1,{_,a:{_,x_,_},_,b:{_,x_,_},_}:>Union[a,b]]]&,c]
Table[Sum[Length[swell[c]],{c,Select[stableSets[Select[Subsets[Range[n]],Length[#]>1&],Complement[#1,#2]=={}&],And[Union@@#==Range[n],clutQ[#]]&]}],{n,nn}]

A324837 Number of minimal subsets of {1...n} with least common multiple n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 8, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 8, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 16, 1, 2, 3, 1, 2, 8, 1, 3, 2, 8, 1, 7, 1, 2, 3, 3, 2, 8, 1, 5, 1, 2, 1, 16, 2, 2

Offset: 1

Gus Wiseman, Mar 19 2019

nonn

Note that the elements must be pairwise indivisible divisors of n.

Differs from A303838 at positions {1, 180, 210, ...}. For example, a(210) = 49, A303838(210) = 55. - Gus Wiseman, Apr 01 2019

			The a(30) = 8 subsets are: {30}, {2,15}, {3,10}, {5,6}, {6,10}, {6,15}, {10,15}, {2,3,5}.

Gus Wiseman, Table of n, a(n) for n = 1..300

Cf. A000005, A006126, A074971, A076078, A085945, A285572, A285573, A286518, A286520, A303837, A303838, A324837.

minim[s_]:=Complement[s,First/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
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[minim[Select[Rest[stableSets[Divisors[n],Divisible]],LCM@@#==n&]]],{n,100}]

A320798 Number of non-isomorphic weight-n connected antichains of non-constant multisets with multiset density -1.

Original entry on oeis.org

0, 1, 2, 5, 9, 24, 51, 134, 328, 868

Offset: 1

Gus Wiseman, Nov 29 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 24 multiset partitions:
  {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}
          {{123}}  {{1222}}    {{12222}}    {{112222}}
                   {{1233}}    {{12233}}    {{112233}}
                   {{1234}}    {{12333}}    {{122222}}
                   {{13}{23}}  {{12344}}    {{122333}}
                               {{12345}}    {{123333}}
                               {{12}{233}}  {{123344}}
                               {{13}{233}}  {{123444}}
                               {{14}{234}}  {{123455}}
                                            {{123456}}
                                            {{112}{233}}
                                            {{122}{233}}
                                            {{12}{2333}}
                                            {{123}{344}}
                                            {{124}{344}}
                                            {{125}{345}}
                                            {{13}{2233}}
                                            {{13}{2333}}
                                            {{13}{2344}}
                                            {{133}{233}}
                                            {{14}{2344}}
                                            {{15}{2345}}
                                            {{13}{24}{34}}
                                            {{14}{24}{34}}

Cf. A007716, A007718, A286520, A304867, A319719, A319721, A321155, A321760, A322111, A322113.

A320275 Numbers whose distinct prime indices are 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, 9, 13, 19, 27, 37, 49, 53, 61, 81, 89, 91, 113, 131, 151, 169, 223, 243, 247, 251, 281, 299, 311, 343, 359, 361, 377, 427, 463, 503, 593, 611, 637, 659, 689, 703, 719, 729, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1183, 1291, 1321, 1339

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 connected 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}}
    9: {{1},{1}}
   13: {{1,2}}
   19: {{1,1,1}}
   27: {{1},{1},{1}}
   37: {{1,1,2}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   81: {{1},{1},{1},{1}}
   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}}
  169: {{1,2},{1,2}}

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A286520, A290103, A293994, A302242, A316476, A319496, A319837, 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[normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==1]&]

cp's OEIS Frontend

A303838 Number of z-forests with least common multiple n > 1.

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A218970 Number of connected cyclic conjugacy classes of subgroups of the symmetric group.

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A304717 Number of connected strict integer partitions of n with pairwise indivisible parts.

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A305193 Number of connected factorizations of n.

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A304382 Number of z-trees summing to n. Number of connected strict integer partitions of n with pairwise indivisible parts and clutter density -1.

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A317634 Number of caps (also clutter partitions) of clutters (connected antichains) spanning n vertices.

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A317635 Number of connected vertex sets of clutters (connected antichains) spanning n vertices.

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A324837 Number of minimal subsets of {1...n} with least common multiple n.

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