A367911 -id:A367911 - cp's OEIS frontend

A367905 Number of ways to choose a sequence of different binary indices, one of each binary index of n.

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

Offset: 0

Gus Wiseman, Dec 10 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}.

			352 has binary indices of binary indices {{2,3},{1,2,3},{1,4}}, and there are six possible choices (2,1,4), (2,3,1), (2,3,4), (3,1,4), (3,2,1), (3,2,4), so a(352) = 6.

John Tyler Rascoe, Table of n, a(n) for n = 0..16384
Wikipedia, Axiom of choice.

A version for multisets is A367771, see A355529, A355740, A355744, A355745.
Positions of positive terms are A367906.
Positions of zeros are A367907.
Positions of ones are A367908.
Positions of terms > 1 are A367909.
Positions of first appearances are A367910, sorted A367911.
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. A000612, A055621, A072639, A309326, A326031, A326675, A326702, A326753, A367902, A367903, A367904, A367912.
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];
Table[Length[Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]],{n,0,100}]

from itertools import count, islice, product
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen(): #generator of terms
    for n in count(0):
        c = 0
        for j in list(product(*[bin_i(k) for k in bin_i(n)])):
            if len(set(j)) == len(j):
                c += 1
        yield c
A367905_list = list(islice(a_gen(), 90)) # John Tyler Rascoe, May 22 2024

A367907 Numbers n such that it is not possible to choose a different binary index of each binary index of n.

Original entry on oeis.org

7, 15, 23, 25, 27, 29, 30, 31, 39, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121

Offset: 1

Gus Wiseman, Dec 11 2023

Also BII-numbers of set-systems (sets of nonempty sets) contradicting a strict version of the axiom of choice.
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},{2},{1,2},{1,3}} with BII-number 23 has choices (1,2,1,1), (1,2,1,3), (1,2,2,1), (1,2,2,3), but none of these has all different elements, so 23 is in the sequence.
The terms together with the corresponding set-systems begin:
   7: {{1},{2},{1,2}}
  15: {{1},{2},{1,2},{3}}
  23: {{1},{2},{1,2},{1,3}}
  25: {{1},{3},{1,3}}
  27: {{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}}
  39: {{1},{2},{1,2},{2,3}}
  42: {{2},{3},{2,3}}
  43: {{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}}
  51: {{1},{2},{1,3},{2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
Wikipedia, Axiom of choice.

These set-systems are counted by A367903, non-isomorphic A368094.
Positions of zeros in A367905, firsts A367910, sorted A367911.
The complement is A367906.
If there is one unique choice we get A367908, counted by A367904.
If there are multiple choices we get A367909, counted by A367772.
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.
Cf. A000612, A055621, A072639, A083323, A309326, A326702, A326753, A367769, A367901, A367902, A367912.
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], Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]=={}&]

from itertools import count, islice, product
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen(): #generator of terms
    for n in count(1):
        p = list(product(*[bin_i(k) for k in bin_i(n)]))
        x = len(p)
        for j in range(x):
            if len(set(p[j])) == len(p[j]): break
            if j+1 == x: yield(n)
A367907_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Feb 10 2024

A367907 U A367908 U A367909 = A000027.

A367906 Numbers k such that it is possible to choose a different binary index of each binary index of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 48, 49, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 80, 81, 82, 84, 88, 96, 97, 98, 100, 104, 112, 128, 129, 130, 131, 132

Offset: 1

Gus Wiseman, Dec 11 2023

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice.
A binary index of k (row k 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 k to be obtained by taking the binary indices of each binary index of k. 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 {{2,3},{1,2,3},{1,4}} with BII-number 352 has choices such as (2,1,4) that satisfy the axiom, so 352 is in the sequence.
The terms together with the corresponding set-systems begin:
   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}}
  16: {{1,3}}
  17: {{1},{1,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
Wikipedia, Axiom of choice.

These set-systems are counted by A367902, non-isomorphic A368095.
Positions of positive terms in A367905, firsts A367910, sorted A367911.
The complement is A367907.
If there is one unique choice we get A367908, counted by A367904.
If there are multiple choices we get A367909, counted by A367772.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A055621, A059519, A072639, A083323, A309326, A326702, A326753, A367770, A367912.
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], Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]!={}&]

from itertools import count, islice, product
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen(): #generator of terms
    for n in count(1):
        for j in list(product(*[bin_i(k) for k in bin_i(n)])):
            if len(set(j)) == len(j):
                yield(n); break
A367906_list = list(islice(a_gen(),100)) # John Tyler Rascoe, Dec 23 2023

A367908 Numbers n such that there is only one way to choose a different binary index of each binary index of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 56, 67, 69, 70, 73, 74, 81, 88, 98, 104, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152, 154, 156, 162, 163, 165, 166, 168

Offset: 1

Gus Wiseman, Dec 11 2023

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in exactly 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 exactly one way, namely (1,2,3), so 21 is in the sequence.
The terms together with the corresponding set-systems begin:
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..2000
Wikipedia, Axiom of choice.

These set-systems are counted by A367904.
Positions of 1's 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 are multiple choices we get A367909, counted by A367772.
The version for MM-numbers of multiset partitions is A368101.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A059201 counts covering T_0 set-systems.
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 for axiom, complement A368097.
Cf. A000612, A059519, A309326, A326675, A326702, A326753, A326872, A355529, A367902, A367912.
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&]

from itertools import count, islice, product
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen(): #generator of terms
    for n in count(1):
        p = list(product(*[bin_i(k) for k in bin_i(n)]))
        x,c = len(p),0
        for j in range(x):
            if len(set(p[j])) == len(p[j]): c += 1
            if j+1 == x and c == 1: yield(n)
A367908_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Feb 10 2024

A367907 U A367908 U A367909 = A000027.

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

Original entry on oeis.org

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.

A367910 Least number k such that there are exactly n ways to choose a different binary index of each binary index of k.

Original entry on oeis.org

7, 1, 4, 20, 68, 320, 352, 1088, 3136, 13376, 16704, 5184, 82240, 70720, 17472

Offset: 0

Gus Wiseman, Dec 16 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}}
     68: {{1,2},{1,2,3}}
    320: {{1,2,3},{1,4}}
    352: {{2,3},{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
   3136: {{1,2,3},{1,2,4},{3,4}}
  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  16704: {{1,2,3},{1,4},{1,2,3,4}}
   5184: {{1,2,3},{1,2,4},{1,3,4}}
  82240: {{1,2,3},{1,4},{1,2,3,4},{1,5}}
  70720: {{1,2,3},{1,2,4},{1,3,4},{1,5}}

Wikipedia, Axiom of choice.

Positions of first appearances in A367905.
The sorted version is A367911.
For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.
Not requiring distinctness gives A368111, firsts of A368109, sorted A368112.
For multisets of indices we have A368184, firsts of A368183, 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, A367771, A367906, A367907, A367908, A367909.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]],{n,1000}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[c,n][[1,1]],{n,0,spnm[c]}]

A367913 Least number k such that there are exactly n ways to choose a multiset consisting of a binary index of each binary index of k.

Original entry on oeis.org

1, 4, 64, 20, 68, 320, 52, 84, 16448, 324, 832, 116, 1104, 308, 816, 340, 836, 848, 1108, 1136, 1360, 3152, 16708, 372, 5188, 5216, 852, 880, 2884, 1364, 13376, 1392, 3184, 3424, 17220, 5204, 5220, 2868, 5728, 884, 19536, 66896, 2900, 1396, 21572, 3188, 3412

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}}
     64: {{1,2,3}}
     20: {{1,2},{1,3}}
     68: {{1,2},{1,2,3}}
    320: {{1,2,3},{1,4}}
     52: {{1,2},{1,3},{2,3}}
     84: {{1,2},{1,3},{1,2,3}}
  16448: {{1,2,3},{1,2,3,4}}
    324: {{1,2},{1,2,3},{1,4}}
    832: {{1,2,3},{1,4},{2,4}}
    116: {{1,2},{1,3},{2,3},{1,2,3}}

A version for multisets and divisors is A355734.
With distinctness we have A367910, firsts of A367905, sorted A367911.
Positions of first appearances in A367912.
The sorted version is A367915.
For sequences we have A368111, firsts of A368109, sorted A368112.
For sets we have A368184, firsts of A368183, 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, A326749, A326753, A355733, A355741, A355744.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]],{n,1000}];
Table[Position[c,n][[1,1]],{n,spnm[c]}]

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

Original entry on oeis.org

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[[#]]]&]

cp's OEIS Frontend

A367905 Number of ways to choose a sequence of different binary indices, one of each binary index of n.

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A367907 Numbers n such that it is not possible to choose a different binary index of each binary index of n.

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A367906 Numbers k such that it is possible to choose a different binary index of each binary index of k.

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A367908 Numbers n such that there is only one way to choose a different binary index of each binary index of n.

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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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A367910 Least number k such that there are exactly n ways to choose a different binary index of each binary index of k.

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A367913 Least number k such that there are exactly n ways to choose a multiset consisting of a binary index of each binary index of k.

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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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