A326749 -id:A326749 - cp's OEIS frontend

A327111 BII-numbers of set-systems with spanning edge-connectivity 1.

1, 2, 4, 5, 6, 7, 8, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 88, 89, 90, 91, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Aug 25 2019

nonn

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) 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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-connectivity 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  32: {{2,3}}

Graphs with spanning edge-connectivity >= 2 are counted by A095983.
BII-numbers for vertex-connectivity 1 are A327098.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
Graphs with spanning edge-connectivity 1 are counted by A327145.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Cf. A013922, A322395, A326749, A327041, A327069, A327071, A327097, A327144, A327145.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]

A326787 Non-spanning edge-connectivity of the set-system with BII-number n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 25 2019

nonn

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 finite set of finite nonempty sets 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 non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.

			Positions of first appearances of each integer together with the corresponding set-systems:
     0: {}
     1: {{1}}
     5: {{1},{1,2}}
    21: {{1},{1,2},{1,3}}
    85: {{1},{1,2},{1,3},{1,2,3}}
   341: {{1},{1,2},{1,3},{1,4},{1,2,3}}
  1365: {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4}}
  5461: {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4},{1,3,4}}

Wikipedia, k-edge-connected graph

Cf. A000120, A013922, A048793, A070939, A095983, A322336, A322338 (same for MM-numbers), A326031, A326749, A326753, A326786 (vertex-connectivity).

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]]]]]]]]];
eConn[sys_]:=Length[sys]-Max@@Length/@Select[Subsets[sys],Length[csm[#]]!=1&];
Table[eConn[bpe/@bpe[n]],{n,0,100}]

A327108 BII-numbers of set-systems with spanning edge-connectivity 2.

Original entry on oeis.org

52, 53, 54, 55, 60, 61, 62, 63, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113, 114, 115, 120, 121, 122, 123, 772, 773, 774, 775, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 848, 849, 850

Offset: 1

Gus Wiseman, Aug 23 2019

nonn

Differs from A327109 in lacking 116, 117, 118, 119, 124, 125, 126, 127, ...
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) 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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-connectivity 2 together with their BII-numbers begins:
   52: {{1,2},{1,3},{2,3}}
   53: {{1},{1,2},{1,3},{2,3}}
   54: {{2},{1,2},{1,3},{2,3}}
   55: {{1},{2},{1,2},{1,3},{2,3}}
   60: {{1,2},{3},{1,3},{2,3}}
   61: {{1},{1,2},{3},{1,3},{2,3}}
   62: {{2},{1,2},{3},{1,3},{2,3}}
   63: {{1},{2},{1,2},{3},{1,3},{2,3}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   86: {{2},{1,2},{1,3},{1,2,3}}
   87: {{1},{2},{1,2},{1,3},{1,2,3}}
   92: {{1,2},{3},{1,3},{1,2,3}}
   93: {{1},{1,2},{3},{1,3},{1,2,3}}
   94: {{2},{1,2},{3},{1,3},{1,2,3}}
   95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  101: {{1},{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}
  103: {{1},{2},{1,2},{2,3},{1,2,3}}

Positions of 2's in A327144.
Graphs with spanning edge-connectivity >= 2 are counted by A095983.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A326749, A326787, A327041, A327069, A327075, A327102, A327103, A327104.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==2&]

A309326 BII-numbers of minimal covers.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 320, 512, 513, 516, 520, 521, 524, 528, 544, 545, 548

Offset: 1

Gus Wiseman, Jul 23 2019

nonn

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 finite set of finite nonempty sets 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 minimal cover is a set-system where every edge contains at least one vertex that does not belong to any other edge.

			The sequence of all minimal covers together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    4: {{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   12: {{1,2},{3}}
   16: {{1,3}}
   18: {{2},{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   33: {{1},{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}

Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],And@@Table[Union@@Delete[bpe/@bpe[#],i]!=Union@@bpe/@bpe[#],{i,Length[bpe/@bpe[#]]}]&]

A326751 BII-numbers of blobs.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 52, 64, 128, 256, 512, 772, 816, 820, 832, 1024, 1072, 1088, 2048, 2320, 2340, 2356, 2368, 2580, 2592, 2612, 2624, 2836, 2852, 2864, 2868, 2880, 3088, 3104, 3120, 3136, 4096, 4132, 4160, 4612, 4640, 4644, 4672, 5120, 5152, 5184, 8192

Offset: 1

Gus Wiseman, Jul 23 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets 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. In an antichain, no edge is a subset or superset of any other edge. In a 2-vertex-connected set-system, at least two vertices must be removed to make the set-system disconnected. A blob is a connected, 2-vertex-connected antichain of finite, nonempty sets, or, equivalently, a 2-vertex-connected clutter.

			The sequence of all blobs together with their BII-numbers begins:
     0: {}
     1: {{1}}
     2: {{2}}
     4: {{1,2}}
     8: {{3}}
    16: {{1,3}}
    32: {{2,3}}
    52: {{1,2},{1,3},{2,3}}
    64: {{1,2,3}}
   128: {{4}}
   256: {{1,4}}
   512: {{2,4}}
   772: {{1,2},{1,4},{2,4}}
   816: {{1,3},{2,3},{1,4},{2,4}}
   820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
   832: {{1,2,3},{1,4},{2,4}}
  1024: {{1,2,4}}
  1072: {{1,3},{2,3},{1,2,4}}
  1088: {{1,2,3},{1,2,4}}
  2048: {{3,4}}
  2320: {{1,3},{1,4},{3,4}}
  2340: {{1,2},{2,3},{1,4},{3,4}}
  2356: {{1,2},{1,3},{2,3},{1,4},{3,4}}

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

Cf. A000120, A002218, A013922 (2-vertex-connected graphs), A030019, A048143 (clutters), A048793, A070939, A095983, A275307 (spanning blobs), A304118, A304887, A322117, A322397 (2-edge-connected clutters), A326031.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326752 (hypertrees), A326754 (covers).

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}];
tvcQ[eds_]:=And@@Table[Length[csm[DeleteCases[eds,i,{2}]]]<=1,{i,Union@@eds}];
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]]]]]]]]];
Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&&tvcQ[bpe/@bpe[#]]&]

A326872 BII-numbers of connectedness systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26, 27, 32, 33, 34, 35, 40, 41, 42, 43, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Jul 29 2019

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.
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 finite set of finite nonempty sets 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 enumeration of these set-systems by number of covered vertices is given by A326870.

			The sequence of all connectedness systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  18: {{2},{1,3}}
  19: {{1},{2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  26: {{2},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  32: {{2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Connectedness systems are counted by A326866, with unlabeled version A326867.
The case without singletons is A326873.
The connected case is A326879.
Set-systems closed under union are counted by A102896, with BII numbers A326875.
Cf. A029931, A048793, A072446, A326031, A326749, A326753, A326870, A326876, A326879.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
connsysQ[eds_]:=SubsetQ[eds,Union@@@Select[Tuples[eds,2],Intersection@@#!={}&]];
Select[Range[0,100],connsysQ[bpe/@bpe[#]]&]

from itertools import count, islice, combinations
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        E,f = [bin_i(k) for k in bin_i(n)],0
        for i in combinations(E,2):
            if list(set(i[0])|set(i[1])) not in E and len(set(i[0])&set(i[1])) > 0:
                f += 1
                break
        if f < 1:
            yield n
A326872_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Mar 07 2025

A327098 BII-numbers of set-systems with cut-connectivity 1.

Original entry on oeis.org

1, 2, 8, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 128, 260, 261, 262, 263, 272, 273, 276, 277, 278, 279, 280, 281, 284, 285, 286, 287, 292, 293, 294, 295, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309

Offset: 1

Gus Wiseman, Aug 21 2019

nonn

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) 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.

We define the cut-connectivity (A326786, A327237), of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex and no edges has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039), this is the same as vertex-connectivity (A327334, A327051).

			The sequence of all set-systems with cut-connectivity 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   8: {{3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}
  38: {{2},{1,2},{2,3}}
  39: {{1},{2},{1,2},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  46: {{2},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  48: {{1,3},{2,3}}

A subset of A326749.
Positions of 1's in A326786.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity 1 are A327111.
Integer partitions with cut-connectivity 1 are counted by A322390.
Labeled connected separable graphs are counted by A327114.
Connected separable set-systems are counted by A327197.
Cf. A000120, A048793, A070939, A322389, A326031, A327100, A327125.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
vertConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Select[Range[0,100],vertConnSys[bpe/@bpe[#]]==1&]

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.

A326752 BII-numbers of hypertrees.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 64, 128, 256, 260, 272, 276, 292, 304, 320, 512, 516, 532, 544, 548, 560, 576, 768, 784, 800, 1024, 1040, 1056, 2048, 2064, 2068, 2080, 2084, 2096, 2112, 2304, 2308, 2336, 2560, 2564, 2576, 2816, 3072, 4096, 4100, 4128, 4608

Offset: 1

Gus Wiseman, Jul 23 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets 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. In an antichain, no edge is a subset or superset of any other edge. A hypertree is a connected antichain of nonempty sets with density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices.

			The sequence of all hypertrees together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    8: {{3}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  256: {{1,4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  292: {{1,2},{2,3},{1,4}}
  304: {{1,3},{2,3},{1,4}}
  320: {{1,2,3},{1,4}}

Cf. A000120, A000272, A030019 (spanning hypertrees), A035053, A048143, A048793, A052888, A070939, A134954, A275307, A326031, A326702, A326753.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326754 (covers).

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}];
density[c_]:=Total[(Length[#1]-1&)/@c]-Length[Union@@c];
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]]]]]]]]];
Select[Range[0,1000],#==0||stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&&density[bpe/@bpe[#]]==-1&]

A327099 BII-numbers of set-systems with non-spanning edge-connectivity 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 16, 22, 23, 25, 28, 29, 30, 31, 32, 37, 39, 42, 44, 45, 46, 47, 49, 50, 51, 57, 58, 59, 64, 67, 73, 74, 75, 76, 77, 78, 79, 82, 83, 90, 91, 97, 99, 105, 107, 128, 256, 262, 263, 278, 279, 280, 281, 284, 285, 286, 287, 292, 293, 294, 295, 300

Offset: 1

Gus Wiseman, Aug 21 2019

nonn

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) 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 non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to result in a disconnected or empty set-system.

			The sequence of all set-systems with non-spanning edge-connectivity 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  25: {{1},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  32: {{2,3}}
  37: {{1},{1,2},{2,3}}
  39: {{1},{2},{1,2},{2,3}}
  42: {{2},{3},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  46: {{2},{1,2},{3},{2,3}}

Positions of 1's in A326787.
Simple graphs with non-spanning edge-connectivity 1 are A327071.
BII-numbers for non-spanning edge-connectivity >= 1 are A326749.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity 1 are A327111.
BII-numbers for vertex-connectivity 1 are A327114.
Covering set-systems with non-spanning edge-connectivity 1 are counted by A327129.
Cf. A048793, A052446, A070939, A322338, A326031, A327108.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
Select[Range[0,100],edgeConn[bpe[#]]==1&]

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A326752 BII-numbers of hypertrees.

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