A326749 -id:A326749 - cp's OEIS frontend

A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

1, 2, 4, 8, 16, 32, 40, 64, 128, 160, 256, 288, 296, 416, 512, 520, 544, 552, 640, 672, 800, 808, 928, 1024, 2048, 2176, 2304, 2432, 2560, 2688, 2816, 2944, 4096, 8192, 8200, 8224, 8232, 8320, 8352, 8480, 8488, 8608, 8704, 8712, 8736, 8744, 8832, 8864, 8992

Offset: 1

Gus Wiseman, Mar 28 2024

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.

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.

			The terms together with their prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    4: {{2}}
    8: {{1,1}}
   16: {{3}}
   32: {{1,2}}
   40: {{1,1},{1,2}}
   64: {{4}}
  128: {{1,1,1}}
  160: {{1,2},{1,1,1}}
  256: {{2,2}}
  288: {{1,2},{2,2}}
  296: {{1,1},{1,2},{2,2}}
  416: {{1,2},{1,1,1},{2,2}}
  512: {{1,3}}
  520: {{1,1},{1,3}}
  544: {{1,2},{1,3}}
  552: {{1,1},{1,2},{1,3}}
  640: {{1,1,1},{1,3}}
  672: {{1,2},{1,1,1},{1,3}}
  800: {{1,2},{2,2},{1,3}}
  808: {{1,1},{1,2},{2,2},{1,3}}
  928: {{1,2},{1,1,1},{2,2},{1,3}}

Connected case of A087086, relatively prime A328671.
For binary indices of binary indices we have A326750, non-primitive A326749.
For prime indices of prime indices we have A329559, non-primitive A305078.
Primitive case of A371291 = positions of ones in A371452.
For binary indices of prime indices we have A371445, non-primitive A325118.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326964 counts connected set-systems, covering A323818.
Cf. A001222, A051026, A285572, A303362, A304713, A305079, A316476, A319496, A319719, A326704, A371446.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],stableQ[bpe[#],Divisible]&&connectedQ[prix/@bpe[#]]&]

Intersection of A087086 and A371291.

A326964 Number of connected set-systems covering a subset of {1..n}.

Original entry on oeis.org

1, 2, 7, 112, 32253, 2147316942, 9223372023968335715, 170141183460469231667123699322514272668, 5789604461865809771178549250434395393752402807429031284280914691514037561273

Offset: 0

Gus Wiseman, Aug 10 2019

nonn

A set-system is a finite set of finite nonempty sets.

			The a(0) = 1 through a(2) = 7 set-systems:
  {}    {}     {}
        {{1}}  {{1}}
               {{2}}
               {{1,2}}
               {{1},{1,2}}
               {{2},{1,2}}
               {{1},{2},{1,2}}

Covering sets of subsets are A000371.
Connected graphs are A001187.
The unlabeled version is A309667.
The BII-numbers of connected set-systems are A326749.
The covering case is A323818.
Cf. A007718, A048143, A058891, A092918, A300913, A304716, A326866, A326948.

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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Length[csm[#]]<=1&]],{n,0,4}]

Binomial transform of A323818.

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&]]}]

A326879 BII-numbers of connected connectedness systems.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 24, 25, 32, 34, 40, 42, 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, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112

Offset: 1

Gus Wiseman, Jul 29 2019

nonn

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. It is connected if it contains an edge containing all the vertices.
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 connected connectedness systems by number of vertices is given by A326868.

			The sequence of all connected connectedness systems together with their BII-numbers begins:
   0: {}
   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}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  67: {{1},{2},{1,2,3}}
  68: {{1,2},{1,2,3}}

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

Connected connectedness systems are counted by A326868, with unlabeled version A326869.
Connected connectedness systems without singletons are counted by A072447.
The not necessarily connected case is A326872.
Cf. A029931, A048793, A072445, A072446, A326031, A326749, A326753, A326866, A326867, A326870, A326876.

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

A327100 BII-numbers of antichains of sets with cut-connectivity 1.

Original entry on oeis.org

1, 2, 8, 20, 36, 48, 128, 260, 272, 276, 292, 304, 308, 320, 516, 532, 544, 548, 560, 564, 576, 768, 784, 788, 800, 804, 1040, 1056, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304, 2308, 2324, 2336, 2352, 2560, 2564, 2576, 2596, 2608, 2816, 2820, 2832, 2848

Offset: 1

Gus Wiseman, Aug 22 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 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 has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).

			The sequence of all antichains of sets with vertex-connectivity 1 together with their BII-numbers begins:
    1: {{1}}
    2: {{2}}
    8: {{3}}
   20: {{1,2},{1,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
  128: {{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}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  320: {{1,2,3},{1,4}}
  516: {{1,2},{2,4}}
  532: {{1,2},{1,3},{2,4}}
  544: {{2,3},{2,4}}
  548: {{1,2},{2,3},{2,4}}
  560: {{1,3},{2,3},{2,4}}
  564: {{1,2},{1,3},{2,3},{2,4}}

Positions of 1's in A326786.
The graphical case is A327114.
BII numbers of antichains with vertex-connectivity >= 1 are A326750.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for cut-connectivity 1 are A327098.
Cf. A000120, A000372, A006126, A048143, A048793, A070939, A322390, A326031, A326749, A326751, A327071, A327111.

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]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
Select[Range[0,100],stableQ[bpe/@bpe[#],SubsetQ]&&cutConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]

If (+) is union and (-) is complement, we have A327100 = A058891 + (A326750 - A326751).

A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments, counted by A371446.
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.
A binary carry of two positive integers is an overlap of binary indices. A multiset is said to be binary carry-connected iff the graph whose vertices are the elements and whose edges are binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The terms together with their prime indices begin:
     2: {1}            37: {12}              97: {25}
     3: {2}            41: {13}             101: {26}
     4: {1,1}          43: {14}             103: {27}
     5: {3}            47: {15}             107: {28}
     7: {4}            49: {4,4}            109: {29}
     8: {1,1,1}        53: {16}             113: {30}
     9: {2,2}          55: {3,5}            115: {3,9}
    11: {5}            59: {17}             121: {5,5}
    13: {6}            61: {18}             125: {3,3,3}
    16: {1,1,1,1}      64: {1,1,1,1,1,1}    127: {31}
    17: {7}            65: {3,6}            128: {1,1,1,1,1,1,1}
    19: {8}            67: {19}             131: {32}
    23: {9}            71: {20}             137: {33}
    25: {3,3}          73: {21}             139: {34}
    27: {2,2,2}        79: {22}             143: {5,6}
    29: {10}           81: {2,2,2,2}        145: {3,10}
    31: {11}           83: {23}             149: {35}
    32: {1,1,1,1,1}    89: {24}             151: {36}

Contains all powers of primes A000961 except 1.
Case of A325118 (counted by A325098) without binary containments.
For binary indices of binary indices we have A326750 = A326704 /\ A326749.
For prime indices of prime indices we have A329559 = A305078 /\ A316476.
An opposite version is A371294 = A087086 /\ A371291.
Partitions of this type are counted by A371446.
Carry-connected case of A371455 (counted by A325109).
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Select[Range[100],stableQ[bpe/@prix[#],SubsetQ] && Length[csm[bpe/@prix[#]]]==1&]

Intersection of A371455 and A325118.

A371451 Number of connected components of the binary indices of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 01 2024

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.

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 binary indices of prime indices of 805 are {{1,2},{3},{1,4}}, with 2 connected components {{1,2},{1,4}} and {{3}}, so a(805) = 2.

For prime indices of prime indices we have A305079, ones A305078.
Positions of ones are A325118.
Positions of first appearances are A325782.
For prime indices of binary indices we have A371452, ones A371291.
For binary indices of binary indices we have A326753, ones A326749.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A326964 counts connected set-systems, covering A323818.
Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.

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]]]]]]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[csm[bix/@prix[n]]],{n,100}]

zero_first_elem_and_bitmask_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2, #ys, if(ys[j]&&(0!=bitand(cs[i], ys[j])), listput(cs, ys[j]); ys[j] = 0)); i++); (ys); };
A371451(n) = if(1==n, 0, my(cs = apply(p -> primepi(p), factor(n)[, 1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_bitmask_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Jan 29 2025

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A326873 BII-numbers of connectedness systems without singletons.

Original entry on oeis.org

0, 4, 16, 32, 64, 68, 80, 84, 96, 100, 112, 116, 256, 288, 512, 528, 1024, 1028, 1280, 1284, 1536, 1540, 1792, 1796, 2048, 2052, 4096, 4112, 4352, 4368, 6144, 6160, 6400, 6416, 8192, 8224, 8704, 8736, 10240, 10272, 10752, 10784, 16384, 16388, 16400, 16416

Offset: 1

Gus Wiseman, Jul 29 2019

nonn

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 A326877.

			The sequence of all connectedness systems without singletons together with their BII-numbers begins:
     0: {}
     4: {{1,2}}
    16: {{1,3}}
    32: {{2,3}}
    64: {{1,2,3}}
    68: {{1,2},{1,2,3}}
    80: {{1,3},{1,2,3}}
    84: {{1,2},{1,3},{1,2,3}}
    96: {{2,3},{1,2,3}}
   100: {{1,2},{2,3},{1,2,3}}
   112: {{1,3},{2,3},{1,2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   256: {{1,4}}
   288: {{2,3},{1,4}}
   512: {{2,4}}
   528: {{1,3},{2,4}}
  1024: {{1,2,4}}
  1028: {{1,2},{1,2,4}}
  1280: {{1,4},{1,2,4}}
  1284: {{1,2},{1,4},{1,2,4}}

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

Connectedness systems without singletons are counted by A072446, with unlabeled case A072444.
Connectedness systems are counted by A326866, with unlabeled case A326867.
BII-numbers of connectedness systems are A326872.
The connected case is A326879.
Cf. A029931, A048793, A072447, A326031, A326749, A326750, A326870, A326875, A326877.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
connnosQ[eds_]:=!MemberQ[Length/@eds,1]&&SubsetQ[eds,Union@@@Select[Tuples[eds,2],Intersection@@#!={}&]];
Select[Range[0,1000],connnosQ[bpe/@bpe[#]]&]

A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 4, 8, 4, 7, 7, 12, 10, 14, 12, 15, 19, 19, 21, 32, 27, 33, 40, 46, 47, 61, 52, 75, 89, 95, 104, 129, 129, 149, 176, 188, 208, 249, 257, 296, 341, 373, 394, 476, 496, 552

Offset: 0

Gus Wiseman, Apr 02 2024

These partitions are ranked by A371445.
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.
A binary carry of two positive integers is an overlap of binary indices. An integer partition is binary carry-connected iff the graph with one vertex for each part and edges corresponding to binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The a(12) = 8 through a(14) = 7 partitions:
  (12)             (13)                         (14)
  (6,6)            (10,3)                       (7,7)
  (9,3)            (5,5,3)                      (9,5)
  (4,4,4)          (1,1,1,1,1,1,1,1,1,1,1,1,1)  (6,5,3)
  (6,3,3)                                       (5,3,3,3)
  (3,3,3,3)                                     (2,2,2,2,2,2,2)
  (2,2,2,2,2,2)                                 (1,1,1,1,1,1,1,1,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)

The first condition (carry-connected) is A325098.
The second condition (stable) is A325109.
Ranks for binary indices of binary indices are A326750 = A326704 /\ A326749.
Ranks for prime indices of prime indices are A329559 = A305078 /\ A316476.
Ranks for prime indices of binary indices are A371294 = A087086 /\ A371291.
Ranks for binary indices of prime indices are A371445 = A325118 /\ A371455.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], stableQ[bix/@Union[#],SubsetQ]&&Length[csm[bix/@#]]<=1&]],{n,0,30}]

A329625 Smallest BII-number of a connected set-system with n edges.

Original entry on oeis.org

0, 1, 5, 7, 23, 31, 63, 127, 383, 511, 1023, 2047, 4095, 8191

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.

			The sequence of terms together with their corresponding set-systems begins:
     0: {}
     1: {{1}}
     5: {{1},{1,2}}
     7: {{1},{2},{1,2}}
    23: {{1},{2},{1,2},{1,3}}
    31: {{1},{2},{1,2},{3},{1,3}}
    63: {{1},{2},{1,2},{3},{1,3},{2,3}}
   127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
   383: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{1,4}}
   511: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4}}

The smallest BII-number of a set-system with n edges is A000225(n).
The smallest BII-number of a set-system with n vertices is A072639(n).
BII-numbers of connected set-systems are A326749.
MM-numbers of connected set-systems are A328514.
The case of clutters is A329627.
Cf. A048143, A048793, A070939, A304716, A305078, A326031, A326753, A329552, A329553, A329626.

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]]]]]]]]];
First/@GatherBy[Select[Range[0,1000],Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]

cp's OEIS Frontend

A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

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A326964 Number of connected set-systems covering a subset of {1..n}.

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

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A326879 BII-numbers of connected connectedness systems.

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A327100 BII-numbers of antichains of sets with cut-connectivity 1.

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A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

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A371451 Number of connected components of the binary indices of the prime indices of n.

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A326873 BII-numbers of connectedness systems without singletons.

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