A326753 -id:A326753 - cp's OEIS frontend

A367915 Sorted positions of first appearances in A367912 (number of multisets that can be obtained by choosing a binary index of each binary index).

1, 4, 20, 52, 64, 68, 84, 116, 308, 320, 324, 340, 372, 816, 832, 836, 848, 852, 880, 884, 1104, 1108, 1136, 1360, 1364, 1392, 1396, 1904, 1908, 2868, 2884, 2900, 2932, 3152, 3184, 3188, 3412, 3424, 3440, 3444, 3952, 3956, 5188, 5204, 5216, 5220, 5236, 5476

Offset: 1

Gus Wiseman, Dec 16 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
     1: {{1}}
     4: {{1,2}}
    20: {{1,2},{1,3}}
    52: {{1,2},{1,3},{2,3}}
    64: {{1,2,3}}
    68: {{1,2},{1,2,3}}
    84: {{1,2},{1,3},{1,2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   308: {{1,2},{1,3},{2,3},{1,4}}
   320: {{1,2,3},{1,4}}
   324: {{1,2},{1,2,3},{1,4}}
   340: {{1,2},{1,3},{1,2,3},{1,4}}
   372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}

A version for multisets and divisors is A355734.
Sorted positions of first appearances in A367912, for sequences A368109.
The unsorted version is A367913.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355744, A367905, A367906, A367911, A368112, A368185.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]],{n,10000}];
Select[Range[Length[c]],FreeQ[Take[c,#-1],c[[#]]]&]

A368111 Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

Original entry on oeis.org

1, 4, 64, 20, 68, 52, 1088, 84, 308, 1092, 116, 5184, 820, 1108, 372, 5188, 2868, 1140, 13376, 884, 5204, 17204, 1396, 13380, 2932, 5236, 275520, 19252, 1908, 13396, 17268, 5492, 275524, 84788, 3956, 13428, 1324096, 19316, 6004, 275540, 215860, 18292, 13684

Offset: 1

Gus Wiseman, Dec 17 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
    1: {{1}}
    4: {{1,2}}
   64: {{1,2,3}}
   20: {{1,2},{1,3}}
   68: {{1,2},{1,2,3}}
   52: {{1,2},{1,3},{2,3}}
   84: {{1,2},{1,3},{1,2,3}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  116: {{1,2},{1,3},{2,3},{1,2,3}}
  820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
  372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}
  884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

With distinctness we have A367910, sorted A367911, firsts of A367905.
For multisets we have A367913, sorted A367915, firsts of A367912.
Positions of first appearances in A368109.
The sorted version is A368112.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A309326, A326031, A326702, A326753, A355731, A355741, A355744, A367906.

nn=10000;
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dd=Select[Range[nn],Max@@First/@FactorInteger[#]<=3&];
qq=Table[Length[Tuples[bpe/@bpe[n]]],{n,nn}];
kk=Select[Range[Length[dd]],SubsetQ[qq,Take[dd,#]]&]
Table[Position[qq,dd[[n]]][[1,1]],{n,kk}]

A368112 Sorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).

Original entry on oeis.org

1, 4, 20, 52, 64, 68, 84, 116, 308, 372, 820, 884, 1088, 1092, 1108, 1140, 1396, 1908, 2868, 2932, 3956, 5184, 5188, 5204, 5236, 5492, 6004, 8052, 13376, 13380, 13396, 13428, 13684, 14196, 16244, 17204, 17268, 18292, 19252, 19316, 20340, 22388, 24436, 30580

Offset: 1

Gus Wiseman, Dec 17 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
    1: {{1}}
    4: {{1,2}}
   20: {{1,2},{1,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
   68: {{1,2},{1,2,3}}
   84: {{1,2},{1,3},{1,2,3}}
  116: {{1,2},{1,3},{2,3},{1,2,3}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}
  820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
  884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

For multisets we have A367915, unsorted A367913, firsts A367912.
Sorted positions of first appearances in A368109.
The unsorted version is A368111.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A326031, A326702, A326753, A355741, A367771, A367905, A367906, A367911, A368184.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Tuples[bpe/@bpe[n]]], {n,1000}];
Select[Range[Length[c]], FreeQ[Take[c,#-1],c[[#]]]&]

A371291 Numbers whose binary indices are connected, where two numbers are connected iff they have a common factor.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 34, 36, 38, 40, 42, 44, 46, 64, 128, 130, 136, 138, 160, 162, 164, 166, 168, 170, 172, 174, 256, 260, 288, 290, 292, 294, 296, 298, 300, 302, 416, 418, 420, 422, 424, 426, 428, 430, 512, 514, 520, 522, 528, 530, 536, 538, 544, 546, 548

Offset: 1

Gus Wiseman, Mar 27 2024

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 empty set is not considered connected.

			The terms together with their binary expansions and binary indices begin:
    1:          1 ~ {1}
    2:         10 ~ {2}
    4:        100 ~ {3}
    8:       1000 ~ {4}
   10:       1010 ~ {2,4}
   16:      10000 ~ {5}
   32:     100000 ~ {6}
   34:     100010 ~ {2,6}
   36:     100100 ~ {3,6}
   38:     100110 ~ {2,3,6}
   40:     101000 ~ {4,6}
   42:     101010 ~ {2,4,6}
   44:     101100 ~ {3,4,6}
   46:     101110 ~ {2,3,4,6}
   64:    1000000 ~ {7}
  128:   10000000 ~ {8}
  130:   10000010 ~ {2,8}
  136:   10001000 ~ {4,8}
  138:   10001010 ~ {2,4,8}
  160:   10100000 ~ {6,8}
  162:   10100010 ~ {2,6,8}
  164:   10100100 ~ {3,6,8}

For prime indices of each prime index we have A305078.
The opposite version is A325118.
For binary indices of each binary index we have A326749.
The pairwise indivisible case is A371294, opposite A371445.
Positions of ones in A371452.
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.
A087086 lists numbers whose binary indices are pairwise indivisible.
A096111 gives product of binary indices.
A326964 counts connected set-systems, covering A323818.
Cf. A000040, A000720, A001222, A305079, A326753, A371446.

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]]]]]]]]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],Length[csm[prix/@bpe[#]]]==1&]

A368183 Number of sets that can be obtained by choosing a different binary index of each binary index of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Dec 17 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,3,2), (2,1,3), both permutations of {1,2,3}, so a(52) = 1.

For sequences we have A367905, firsts A367910, sorted A367911.
Positions of zeros are A367907.
Without distinctness we have A367912, firsts A367913, sorted A367915.
Positions of positive terms are A367906.
For sequences without distinctness: A368109, firsts A368111, sorted A368112.
Positions of first appearances are A368184, sorted A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355739, A355741, A355744, A367771.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]]],{n,0,100}]

A368184 Least k such that there are exactly n ways to choose a set consisting of a different binary index of each binary index of k.

Original entry on oeis.org

7, 1, 4, 20, 276, 320, 1088, 65856, 66112, 66624, 263232

Offset: 0

Gus Wiseman, Dec 18 2023

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
      7: {{1},{2},{1,2}}
      1: {{1}}
      4: {{1,2}}
     20: {{1,2},{1,3}}
    276: {{1,2},{1,3},{1,4}}
    320: {{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
  65856: {{1,2,3},{1,4},{1,5}}
  66112: {{1,2,3},{2,4},{1,5}}
  66624: {{1,2,3},{1,2,4},{1,5}}

For strict sequences: A367910, firsts of A367905, sorted A367911.
For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.
For sequences w/o distinctness: A368111, firsts of A368109, sorted A368112.
Positions of first appearances in A368183.
The sorted version is A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A326031, A326702, A326753, A355739, A355741, A367771, A367906, A367907.

nn=10000;
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]]],{n,nn}];
k=Max@@Select[Range[Max@@q], SubsetQ[q,Range[#]]&]
Table[Position[q,n][[1,1]],{n,0,k}]

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[#]]&]

A368185 Sorted list of positions of first appearances in A368183 (number of sets that can be obtained by choosing a different binary index of each binary index).

Original entry on oeis.org

1, 4, 7, 20, 276, 320, 1088, 65856, 66112, 66624

Offset: 1

Gus Wiseman, Dec 18 2023

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
      1: {{1}}
      4: {{1,2}}
      7: {{1},{2},{1,2}}
     20: {{1,2},{1,3}}
    276: {{1,2},{1,3},{1,4}}
    320: {{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
  65856: {{1,2,3},{1,4},{1,5}}
  66112: {{1,2,3},{2,4},{1,5}}
  66624: {{1,2,3},{1,2,4},{1,5}}

For sequences we have A367911, unsorted A367910, firsts of A367905.
Multisets w/o distinctness: A367915, unsorted A367913, firsts of A367912.
Sequences w/o distinctness: A368112, unsorted A368111, firsts of A368109.
Sorted list of positions of first appearances in A368183.
The unsorted version is A368184.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A326031, A326702, A326753, A355739, A355741, A367771, A367906, A367907.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]]],{n,1000}];
Select[Range[Length[c]], FreeQ[Take[c,#-1],c[[#]]]&]

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

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

A367915 Sorted positions of first appearances in A367912 (number of multisets that can be obtained by choosing a binary index of each binary index).

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A368111 Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

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A368112 Sorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).

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A371291 Numbers whose binary indices are connected, where two numbers are connected iff they have a common factor.

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A368183 Number of sets that can be obtained by choosing a different binary index of each binary index of n.

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A368184 Least k such that there are exactly n ways to choose a set consisting of a different binary index of each binary index of k.

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A326879 BII-numbers of connected connectedness systems.

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A368185 Sorted list of positions of first appearances in A368183 (number of sets that can be obtained by choosing a different binary index of each binary index).

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

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