A286518 -id:A286518 - cp's OEIS frontend

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

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

A328514 MM-numbers of connected sets of sets.

Original entry on oeis.org

1, 2, 3, 5, 11, 13, 17, 29, 31, 39, 41, 43, 47, 59, 65, 67, 73, 79, 83, 87, 101, 109, 113, 127, 129, 137, 139, 149, 157, 163, 167, 179, 181, 191, 195, 199, 211, 233, 235, 237, 241, 257, 269, 271, 277, 283, 293, 303, 313, 317, 319, 331, 339, 347, 349, 353, 365

Offset: 1

Gus Wiseman, Oct 20 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 all connected set of sets together with their MM-numbers begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
  11: {{3}}
  13: {{1,2}}
  17: {{4}}
  29: {{1,3}}
  31: {{5}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  47: {{2,3}}
  59: {{7}}
  65: {{2},{1,2}}
  67: {{8}}
  73: {{2,4}}
  79: {{1,5}}
  83: {{9}}
  87: {{1},{1,3}}

The not-necessarily-connected case is A302494.
BII-numbers of connected set-systems are A326749.
MM-numbers of connected sets of multisets are A328513.
Cf. A005117, A007947, A056239, A112798, A286518, A302242, A302569, A304714, A305078, A305079, A327398.

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]]]]]]]]];
Select[Range[1000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&]

Intersection of A302494 and A305078.

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}]

A322335 Number of 2-edge-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 4, 2, 7, 0, 13, 0, 15, 8, 21, 1, 37, 2, 45, 18, 58, 8, 95, 19, 109, 45, 150, 38, 232, 59, 268, 129, 357, 155, 523, 203, 633, 359, 852, 431, 1185, 609, 1464, 969

Offset: 1

Gus Wiseman, Dec 04 2018

First differs from A108572 at a(17) = 1, A108572(17) = 0.

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part. For example (6,6,3,2) is 2-edge-connected but (6,3,2) is not.

			The a(14) = 15 2-edge-connected integer partitions of 14:
  (7,7)   (6,4,4)   (4,4,4,2)  (4,4,2,2,2)  (4,2,2,2,2,2)  (2,2,2,2,2,2,2)
  (8,6)   (6,6,2)   (6,4,2,2)  (6,2,2,2,2)
  (10,4)  (8,4,2)   (8,2,2,2)
  (12,2)  (10,2,2)

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A322336, A322337, A322338.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Table[Length[Select[IntegerPartitions[n],twoedQ[primeMS/@#]&]],{n,30}]

a(42)-a(45) from Jinyuan Wang, Jun 20 2020

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}]

A322337 Number of strict 2-edge-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 10, 5, 11, 1, 18, 3, 17, 8, 22, 3, 35, 5, 32, 17, 39, 16, 59, 14, 58, 33, 75, 28, 103, 35, 106, 71, 125, 63, 174, 81, 192, 127, 220, 130, 294, 170, 325, 237, 378, 257, 504

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part.

			The a(24) = 18 strict 2-edge-connected integer partitions of 24:
  (15,9)   (10,8,6)   (10,8,4,2)
  (16,8)   (12,8,4)   (12,6,4,2)
  (18,6)   (12,9,3)
  (20,4)   (14,6,4)
  (21,3)   (14,8,2)
  (22,2)   (15,6,3)
  (14,10)  (16,6,2)
           (18,4,2)
           (12,10,2)

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A305078, A305079, A322335, A322336.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,twoedQ[primeMS/@#]]&]],{n,30}]

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}]

A069626 Number of sets of integers larger than one whose least common multiple is n.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 4, 2, 5, 1, 22, 1, 5, 5, 8, 1, 22, 1, 22, 5, 5, 1, 92, 2, 5, 4, 22, 1, 109, 1, 16, 5, 5, 5, 200, 1, 5, 5, 92, 1, 109, 1, 22, 22, 5, 1, 376, 2, 22, 5, 22, 1, 92, 5, 92, 5, 5, 1, 1874, 1, 5, 22, 32, 5, 109, 1, 22, 5, 109, 1, 1696, 1, 5, 22, 22, 5, 109, 1, 376, 8, 5, 1, 1874, 5, 5, 5, 92, 1, 1874, 5, 22

Offset: 1

Amarnath Murthy, Mar 27 2002

a(p) = 1, a(p*q) = 5, a(p^2*q) = 13, a(p^3) = 4, a(p^4) = 8 etc. where p and q are primes. It can be shown that a(p^k) = 2^(k-1). Problem: find an expression for a(N) when N = p^a*q^b*r^c*..., p,q,r are primes.

			a(6) = 5 as there are five such sets of natural numbers larger than one whose least common multiple is six: {6}, {2, 6}, {3, 6}, {2, 3} and {2, 3, 6}.
a(12) = 22 from {12}, {4,3}, {2,4,3}, {4,6}, {2,4,6}, {4,3,6}, {2,4,3,6}, {2,12}, {4,12}, {2,4,12}, {3,12}, {2,3,12}, {4,3,12}, {2,4,3,12}, {6,12}, {2,6,12}, {4,6,12}, {2,4,6,12}, {3,6,12}, {2,3,6,12}, {4,3,6,12}, {2,4,3,6,12}.
From _Antti Karttunen_, Feb 18 2024: (Start)
a(1) = 1 as there is only one set that satisfies the criteria, namely, an empty set {}, whose lcm is 1.
a(2) = 1 as the only set that satisfies the criteria is a singleton set {2}.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences related to lcm's

A000005, A008683, A286518, A318670.
Möbius transform of A100577.
Cf. also A045778 (number of sets of integers > 1 whose product is n).
Cf. A076078.

-- following Vladeta Jovovic's formula.
a069626 n = sum $
   map (\d -> (a008683 (n `div` d)) * 2 ^ (a000005 d - 1)) $ a027750_row n
-- Reinhard Zumkeller, Jun 12 2015, Feb 07 2011
(APL, Dyalog dialect)
divisors ← {ð←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:ð ⋄ ð,(⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽ð}
A069626 ← { D←1↓divisors(⍵) ⋄ T←(⍴D)⍴2 ⋄ +/⍵⍷{∧/D/⍨T⊤⍵}¨(-∘1)⍳2*⍴D } ⍝ (quite taxing on memory) - Antti Karttunen, Feb 18 2024

with(numtheory): seq(add(mobius(n/d)*2^(tau(d)-1), d in divisors(n)), n=1..80); # Ridouane Oudra, Mar 12 2024

a[n_] := Sum[ MoebiusMu[n/d] * 2^(DivisorSigma[0, d] - 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 92}](* Jean-François Alcover, Nov 30 2011, after Vladeta Jovovic *)

A069626(n) = sumdiv(n,d,moebius(n/d)*2^(numdiv(d)-1)); \\ Antti Karttunen, Feb 18 2024

a(n) = Sum_{ d divides n } mu(n/d)*2^(tau(d)-1). - Vladeta Jovovic, Jul 07 2003
a(n) >= A286518, a(n) >= A318670. - Antti Karttunen, Feb 17 2024
a(n) = A076078(n)/2, for n > 1. - Ridouane Oudra, Mar 12 2024

Corrected and extended by Naohiro Nomoto, Apr 25 2002

Definition and examples clarified by Antti Karttunen, Feb 18 2024

A322306 Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 2, 2, 3, 2, 2, 1, 4, 2, 2, 3, 2, 1, 3, 1, 2, 5, 1, 3, 3, 1, 2, 2, 3, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 4, 2, 1, 4, 2, 2, 3

Offset: 1

Gus Wiseman, Dec 03 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A positive integer is connected if its prime indices are connected (see A305078).

			The a(1365) = 12 divisors are 3, 5, 7, 13, 21, 39, 65, 91, 195, 273, 455, 1365. These correspond to the following connected submultisets of {{1},{2},{1,1},{1,2}}.
     3: {{1}}
     5: {{2}}
     7: {{1,1}}
    13: {{1,2}}
    21: {{1},{1,1}}
    39: {{1},{1,2}}
    65: {{2},{1,2}}
    91: {{1,1},{1,2}}
   195: {{1},{2},{1,2}}
   273: {{1},{1,1},{1,2}}
   455: {{2},{1,1},{1,2}}
  1365: {{1},{2},{1,1},{1,2}}

Cf. A003963, A054921, A112798, A286518, A290103, A301957, A302242, A304714, A304716, A305078, A316556, A322307.

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[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[Union[Subsets[primeMS[n]]],Length[zsm[#]]==1&]],{n,50}]

cp's OEIS Frontend

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

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A328514 MM-numbers of connected sets of sets.

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

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A322335 Number of 2-edge-connected integer partitions of n.

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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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A322337 Number of strict 2-edge-connected integer partitions of n.

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

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