A326701 -id:A326701 - cp's OEIS frontend

A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

4, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 65, 66, 68, 72, 76, 80, 82, 84, 96, 97, 100, 112, 132, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 193, 194, 196, 200, 204, 208, 210, 212, 224, 225, 228, 240, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288

Offset: 1

Gus Wiseman, Dec 11 2023

nonn

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in more than one way.
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. 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 digits (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.
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in only one way (1,2,3), so 21 is not in the sequence.
The terms together with the corresponding set-systems begin:
   4: {{1,2}}
  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}}
  52: {{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Wikipedia, Axiom of choice.

These set-systems are counted by A367772.
Positions of terms > 1 in A367905, firsts A367910, sorted firsts A367911.
If there is at least one choice we get A367906, counted by A367902.
If there are no choices we get A367907, counted by A367903.
If there is one unique choice we get A367908, counted by A367904.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A368098 counts unlabeled multiset partitions per axiom, complement A368097.
Cf. A000612, A055621, A072639, A309326, A326702, A326753, A355529, A368100.
BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers), A326783 (uniform), A326784 (regular), A326788 (simple), A330217 (achiral).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100], Length[Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]]>1&]

A367907 U A367908 U A367909 = A000027.

A327105 BII-numbers of set-systems with minimum degree 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 48, 49, 50, 56, 57, 58, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 88, 89, 96, 98, 104, 106, 128

Offset: 1

Gus Wiseman, Aug 26 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.

In a set-system, the degree of a vertex is the number of edges containing it.

			The sequence of all set-systems with minimum degree 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{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}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}

Positions of 1's in A327103.
BII-numbers for minimum degree > 1 are A327107.
Graphs with minimum degree 1 are counted by A245797, with covering case A327227.
Set-systems with minimum degree 1 are counted by A327228, with covering case A327229.
Cf. A000120, A029931, A048793, A058891, A070939, A326031, A326701, A326786, A327041, A327104, A327230.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],If[#==0,0,Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]]==1&]

A326784 BII-numbers of regular set-systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 16, 18, 25, 30, 32, 33, 42, 45, 51, 52, 63, 64, 75, 76, 82, 94, 97, 109, 115, 116, 127, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 160, 161, 192, 256, 258, 264, 266, 288, 385, 390, 408, 427, 428, 434, 458

Offset: 1

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. A set-system is regular if all vertices appear the same number of times.

			The sequence of all regular set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   7: {{1},{2},{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}}
  25: {{1},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  42: {{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}

Cf. A000120, A001511, A005176, A029931, A048793, A070939, A295193, A322554, A326031, A326701, A326783 (uniform), A326785 (uniform regular).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SameQ@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]&]

A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 2, 0, 0, 0, 0, 4, 5, 1, 0, 0, 0, 0, 5, 6, 1, 0, 0, 0, 0, 0, 6, 9, 2, 0, 0, 0, 0, 0, 0, 8, 13, 3, 0, 0, 0, 0, 0, 0, 0, 10, 23, 10, 1, 0, 0, 0, 0, 0, 0, 0, 12, 27, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 15, 40, 19, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 18 2019

			Triangle begins:
  1
  0  1
  0  1  0
  0  2  1  0
  0  2  1  0  0
  0  3  2  0  0  0
  0  4  5  1  0  0  0
  0  5  6  1  0  0  0  0
  0  6  9  2  0  0  0  0  0
  0  8 13  3  0  0  0  0  0  0
  0 10 23 10  1  0  0  0  0  0  0
  0 12 27 11  1  0  0  0  0  0  0  0
  0 15 40 19  2  0  0  0  0  0  0  0  0
Row n = 8 counts the following set partitions:
  {{8}}      {{1},{7}}    {{1},{2},{5}}
  {{3,5}}    {{2},{6}}    {{1},{3},{4}}
  {{2,6}}    {{3},{5}}
  {{1,7}}    {{1},{3,4}}
  {{1,3,4}}  {{1},{2,5}}
  {{1,2,5}}  {{2},{1,5}}
             {{3},{1,4}}
             {{4},{1,3}}
             {{5},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n=0..50)

Row sums are A294617.
Column k = 1 is A000009 (n > 0).
Cf. A000110, A008277, A050342, A060016, A072706, A270995, A271619, A279375, A279785, A326701, A330459, A330462, A330463, A330759.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 k*
         b(n-i, t, k)+b(n-i, t, k+1))(min(n-i, i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..15);  # Alois P. Heinz, Dec 29 2019

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,2],Length[#]==k&&And[UnsameQ@@#,UnsameQ@@Join@@#]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, x^k, b[n, i-1, k] + Function[t, k*b[n-i, t, k] + b[n-i, t, k + 1]][Min[n-i, i-1]]]];
T[n_] := PadRight[CoefficientList[b[n, n, 0], x], n + 1];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n)))); vector(#v, n, my(p=v[n]); vector(n, k, sum(i=k, n, polcoef(p,i-1)*stirling(i-1, k-1, 2))))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

T(n,k) = Sum_{k <= i <= n} A060016(n,i) * A008277(i,k).

For n > 0, T(n,2) = Sum_{k = 1..n} (2^(k - 1) -1) * A060016(n,k).

A327104 Maximum vertex-degree of the set-system with BII-number n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 26 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.

In a set-system, the degree of a vertex is the number of edges containing it.

			The BII-number of {{2},{3},{1,2},{1,3},{2,3}} is 62, and its degrees are (2,3,3), so a(62) = 3.

Positions of 1's are A326701 (BII-numbers of set-partitions).
The minimum vertex-degree is A327103.
Positions of 2's are A327106.
Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326786, A327041.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[If[n==0,0,Max@@Length/@Split[Sort[Join@@bpe/@bpe[n]]]],{n,0,100}]

A326785 BII-numbers of uniform regular set-systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 16, 32, 52, 64, 128, 129, 130, 131, 136, 137, 138, 139, 256, 288, 512, 528, 772, 816, 1024, 2048, 2052, 2320, 2340, 2580, 2592, 2868, 4096, 8192, 13376, 16384, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897

Offset: 1

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. A set-system is uniform if all edges have the same size, and regular if all vertices appear the same number of times.

			The sequence of all uniform regular set-systems 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}}
   16: {{1,3}}
   32: {{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}
  130: {{2},{4}}
  131: {{1},{2},{4}}
  136: {{3},{4}}
  137: {{1},{3},{4}}
  138: {{2},{3},{4}}

Cf. A000120, A029931, A048793, A070939, A319056, A319189, A321698, A326031, A326701, A326783 (uniform), A326784 (regular), A326788.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],SameQ@@Length/@bpe/@bpe[#]&&SameQ@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]&]

Intersection of A326783 and A326784.

A327107 BII-numbers of set-systems with minimum vertex-degree > 1.

Original entry on oeis.org

7, 25, 30, 31, 42, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 99, 100, 101, 102, 103, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

Offset: 1

Gus Wiseman, Aug 26 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.

In a set-system, the degree of a vertex is the number of edges containing it.

			The sequence of all set-systems with maximum degree > 1 together with their BII-numbers begins:
   7: {{1},{2},{1,2}}
  25: {{1},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  42: {{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  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}}
  59: {{1},{2},{3},{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}}
  75: {{1},{2},{3},{1,2,3}}
  76: {{1,2},{3},{1,2,3}}

Positions of terms > 1 in A327103.
BII-numbers for minimum degree 1 are A327105.
Graphs with minimum degree > 1 are counted by A059167.
Cf. A000120, A029931, A048793, A058891, A070939, A245797, A326031, A326701, A326786, A327041, A327104, A327227-A327230.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]>1&]

A329560 BII-numbers of antichains of sets with empty intersection.

Original entry on oeis.org

0, 3, 9, 10, 11, 12, 18, 33, 52, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 258, 264, 266, 268, 274, 288, 292, 304, 308, 513, 520, 521, 524, 528, 532, 545, 560, 564, 772, 776, 780, 784, 788, 800, 804, 816, 820, 832

Offset: 1

Gus Wiseman, Nov 28 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 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.
A set-system is an antichain if no edge is a proper subset of any other.
Empty intersection means there is no vertex in common to all the edges

			The sequence of terms together with their binary expansions and corresponding set-systems begins:
    0:          0 ~ {}
    3:         11 ~ {{1},{2}}
    9:       1001 ~ {{1},{3}}
   10:       1010 ~ {{2},{3}}
   11:       1011 ~ {{1},{2},{3}}
   12:       1100 ~ {{1,2},{3}}
   18:      10010 ~ {{2},{1,3}}
   33:     100001 ~ {{1},{2,3}}
   52:     110100 ~ {{1,2},{1,3},{2,3}}
  129:   10000001 ~ {{1},{4}}
  130:   10000010 ~ {{2},{4}}
  131:   10000011 ~ {{1},{2},{4}}
  132:   10000100 ~ {{1,2},{4}}
  136:   10001000 ~ {{3},{4}}
  137:   10001001 ~ {{1},{3},{4}}
  138:   10001010 ~ {{2},{3},{4}}
  139:   10001011 ~ {{2},{3},{4}}
  140:   10001100 ~ {{1,2},{3},{4}}
  144:   10010000 ~ {{1,3},{4}}
  146:   10010010 ~ {{2},{1,3},{4}}
  148:   10010100 ~ {{1,2},{1,3},{4}}

Intersection of A326911 and A326704.
BII-numbers of intersecting set-systems with empty intersecting are A326912.
Cf. A000120, A048793, A070939, A326031, A326701, A328671, A329561, A329626, A329628, A329661.

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}];
Select[Range[0,100],#==0||Intersection@@bpe/@bpe[#]=={}&&stableQ[bpe/@bpe[#],SubsetQ]&]

A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.

Original entry on oeis.org

1, 2, 3, 6, 7, 22, 23, 32, 33, 48, 49, 86, 87, 112, 113, 516, 517, 580, 581, 1110, 1111, 1136, 1137, 1604, 1605, 5206, 5207, 5232, 5233, 5700, 5701, 8212, 8213, 9236, 9237, 13332, 13333, 16386, 16387, 16450, 16451, 17474, 17475, 21570, 21571, 24576, 24577

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}}
    3: {{},{1}}
    6: {{1},{2}}
    7: {{},{1},{2}}
   22: {{1},{2},{3}}
   23: {{},{1},{2},{3}}
   32: {{1,2}}
   33: {{},{1,2}}
   48: {{3},{1,2}}
   49: {{},{3},{1,2}}
   86: {{1},{2},{3},{4}}
   87: {{},{1},{2},{3},{4}}
  112: {{3},{1,2},{4}}
  113: {{},{3},{1,2},{4}}
  516: {{2},{1,3}}
  517: {{},{2},{1,3}}
  580: {{2},{4},{1,3}}
  581: {{},{2},{4},{1,3}}

Without the covering condition we have A371289.
Without squarefree product we have A371292.
Interchanging binary and prime indices gives A371448.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts ordered set partitions, allowing empty sets A000629.
A005117 lists squarefree numbers.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A131689 counts patterns by number of distinct parts.
A302521 lists MM-numbers of set partitions, with empties A302505.
A326701 lists BII-numbers of set partitions.
A368533 lists numbers with squarefree binary indices, prime indices A302478.
Cf. A000040, A001222, A255906, A326782, A371291, A371294, A371447, A371452.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
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[1000],SquareFreeQ[Times @@ bpe[#]]&&normQ[Join@@prix/@bpe[#]]&]

Intersection of A371292 and A371289.

A329661 BII-number of the set-system whose MM-number is A329629(n).

Original entry on oeis.org

0, 1, 2, 8, 4, 3, 128, 16, 32768, 9, 5, 2147483648, 256, 32, 129, 10, 9223372036854775808, 6, 170141183460469231731687303715884105728, 512, 65536, 57896044618658097711785492504343953926634992332820282019728792003956564819968, 130, 17, 32769, 4294967296

Offset: 1

Gus Wiseman, Nov 19 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 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.

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 all set-systems together with their MM-numbers and BII-numbers begins:
             {}:  1 ~ 0
          {{1}}:  3 ~ 1
          {{2}}:  5 ~ 2
          {{3}}: 11 ~ 8
        {{1,2}}: 13 ~ 4
      {{1},{2}}: 15 ~ 3
          {{4}}: 17 ~ 128
        {{1,3}}: 29 ~ 16
          {{5}}: 31 ~ 32768
      {{1},{3}}: 33 ~ 9
    {{1},{1,2}}: 39 ~ 5
          {{6}}: 41 ~ 2147483648
        {{1,4}}: 43 ~ 256
        {{2,3}}: 47 ~ 32
      {{1},{4}}: 51 ~ 129
      {{2},{3}}: 55 ~ 10
          {{7}}: 59 ~ 9223372036854775808
    {{2},{1,2}}: 65 ~ 6
          {{8}}: 67 ~ 170141183460469231731687303715884105728
        {{2,4}}: 73 ~ 512

MM-numbers of set-systems are A329629.
Cf. A000120, A005117, A048793, A056239, A070939, A112798, A302242, A302494, A326031, A329557.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).
Classes of BII-numbers: A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326752 (hypertrees), A326754 (covers).

fbi[q_]:=If[q=={},0,Total[2^q]/2];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
das=Select[Range[100],OddQ[#]&&SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&];
Table[fbi[fbi/@primeMS/@primeMS[n]],{n,das}]

A326031(a(n)) = A302242(A329629(n)).

cp's OEIS Frontend

A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

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A327105 BII-numbers of set-systems with minimum degree 1.

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A326784 BII-numbers of regular set-systems.

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A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

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A327104 Maximum vertex-degree of the set-system with BII-number n.

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A326785 BII-numbers of uniform regular set-systems.

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A327107 BII-numbers of set-systems with minimum vertex-degree > 1.

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A329560 BII-numbers of antichains of sets with empty intersection.

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A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.