A326754 -id:A326754 - 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.

A340020 MM-numbers of labeled graphs with loops, without isolated vertices.

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 553, 559, 577, 607, 611, 631, 647, 653, 661, 667, 673, 677

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a loop is an edge with two equal vertices, distinguished from a half-loop, which has only one vertex.
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}}.
Also products of distinct primes whose prime indices are semiprimes, where a semiprime (A001358) is a product of any two prime numbers.

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
      1: {}              161: {{1,1},{2,2}}    347: {{2,9}}
      7: {{1,1}}         163: {{1,8}}          373: {{1,12}}
     13: {{1,2}}         167: {{2,6}}          377: {{1,2},{1,3}}
     23: {{2,2}}         199: {{1,9}}          389: {{4,5}}
     29: {{1,3}}         203: {{1,1},{1,3}}    421: {{1,13}}
     43: {{1,4}}         227: {{4,4}}          439: {{3,7}}
     47: {{2,3}}         233: {{2,7}}          443: {{1,14}}
     73: {{2,4}}         257: {{3,5}}          449: {{2,10}}
     79: {{1,5}}         269: {{2,8}}          467: {{4,6}}
     91: {{1,1},{1,2}}   271: {{1,10}}         487: {{2,11}}
     97: {{3,3}}         293: {{1,11}}         491: {{1,15}}
    101: {{1,6}}         299: {{1,2},{2,2}}    499: {{3,8}}
    137: {{2,5}}         301: {{1,1},{1,4}}    511: {{1,1},{2,4}}
    139: {{1,7}}         313: {{3,6}}          553: {{1,1},{1,5}}
    149: {{3,4}}         329: {{1,1},{2,3}}    559: {{1,2},{1,4}}

The case with only one edge is A106349.
The case covering an initial interval is A320461.
The version allowing multiple edges is A339112.
The half-loop version covering an initial interval is A340018.
The half-loop version is A340019.
A006450 lists primes of prime index.
A302242 is the weight of the multiset of multisets with MM-number n.
A302494 lists MM-numbers of sets of sets, with connected case A328514.
A309356 lists MM-numbers of simple graphs.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

Select[Range[100],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeOmega[PrimePi[p]]!=2]&]

A330102 BII-number of the VDD-normalization of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 5, 7, 1, 3, 3, 11, 33, 19, 19, 15, 4, 5, 33, 19, 20, 21, 37, 23, 5, 7, 19, 15, 37, 23, 51, 31, 4, 33, 5, 19, 20, 37, 21, 23, 5, 19, 7, 15, 37, 51, 23, 31, 20, 37, 37, 51, 52, 53, 53, 55, 21, 23, 23, 31, 53, 55, 55, 63, 64, 65, 65, 67, 68, 69, 69

Offset: 0

Gus Wiseman, Dec 04 2019

nonn

First differs from A330101 at a(148) = 274, A330101(148) = 545, with corresponding set-systems 274: {{2},{1,3},{1,4}} and 545: {{1},{2,3},{2,4}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the VDD (vertex-degrees decreasing) normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of sets is first by length and then lexicographically.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}
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 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.

			56 is the BII-number of {{3},{1,3},{2,3}}, which has VDD-normalization {{1},{1,2},{1,3}} with BII-number 21, so a(56) = 21.

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330100.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A326754, A330061, A330101.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[fbi[fbi/@sysnorm[bpe/@bpe[n]]],{n,0,100}]

A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

Original entry on oeis.org

0, 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, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152

Offset: 1

Gus Wiseman, Dec 12 2023

nonn

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 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.

			The terms together with the corresponding set-systems begin:
   0: {}
   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}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}

These set-systems are counted by A054780 and A367916, A368186.
Graphs of this type are A367862, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, connected A323818, unlabeled A000612.
A070939 gives length of binary expansion.
A136556 counts set-systems on {1..n} with n edges.
Cf. A057500, A059201, A072639, A096111, A116508, A309326, A326031, A326702, A326753, A326754, A367770, A367902, A367905.

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

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195

Offset: 0

Gus Wiseman, Dec 19 2023

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

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

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

A340018 MM-numbers of labeled graphs with half-loops covering an initial interval of positive integers, without isolated vertices.

Original entry on oeis.org

1, 3, 13, 15, 39, 65, 141, 143, 145, 165, 195, 377, 429, 435, 611, 705, 715, 1131, 1363, 1551, 1595, 1833, 1885, 1937, 2021, 2117, 2145, 2235, 2365, 2397, 2409, 2431, 2465, 2805, 3055, 4089, 4147, 4785, 5655, 5811, 6063, 6149, 6235, 6351, 6409, 6721, 6815

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal vertices.
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}}.
Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime, and whose prime indices together cover an initial interval of positive integers. A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
     1: {}
     3: {{1}}
    13: {{1,2}}
    15: {{1},{2}}
    39: {{1},{1,2}}
    65: {{2},{1,2}}
   141: {{1},{2,3}}
   143: {{3},{1,2}}
   145: {{2},{1,3}}
   165: {{1},{2},{3}}
   195: {{1},{2},{1,2}}
   377: {{1,2},{1,3}}
   429: {{1},{3},{1,2}}
   435: {{1},{2},{1,3}}
   611: {{1,2},{2,3}}
   705: {{1},{2},{2,3}}
   715: {{2},{3},{1,2}}
  1131: {{1},{1,2},{1,3}}

The version with full loops is A320461.
The version not necessarily covering an initial interval is A340019.
MM-numbers of graphs with loops are A340020.
A006450 lists primes of prime index.
A106349 lists primes of semiprime index.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A302494 lists MM-numbers of sets of sets, with connected case A328514.
A309356 lists MM-numbers of simple graphs.
A322551 lists primes of squarefree semiprime index.
A339112 lists MM-numbers of multigraphs with loops.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]

A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 18, 20, 24, 25, 26, 30, 32, 33, 34, 35, 36, 40, 42, 45, 47, 48, 50, 51, 52, 54, 55, 60, 64, 65, 66, 68, 70, 72, 75, 78, 80, 84, 85, 86, 90, 94, 96, 99, 100, 102, 104, 105, 108, 110, 119, 120, 123, 125, 126, 127, 128, 130

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have binary indices covering an initial interval.
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 terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  18: {{1},{2},{2}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  25: {{1,2},{1,2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}

For prime indices of prime indices we have A320456.
For binary indices of binary indices we have A326754.
An opposite version is A371292, A371293.
The case with squarefree product of prime indices is A371448.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
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.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

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

A330195 BII-number of the BII-normalization of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 5, 7, 1, 3, 3, 11, 12, 13, 13, 15, 4, 5, 12, 13, 20, 21, 22, 23, 5, 7, 13, 15, 22, 23, 30, 31, 4, 12, 5, 13, 20, 22, 21, 23, 5, 13, 7, 15, 22, 30, 23, 31, 20, 22, 22, 30, 52, 53, 53, 55, 21, 23, 23, 31, 53, 55, 55, 63, 64, 65, 65, 67, 68, 69

Offset: 0

Gus Wiseman, Dec 05 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 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 BII-normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest BII-number.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330109.
A subset of A326754.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
BII-weight is A326031.
Cf. A000120, A000612, A007716, A048793, A070939, A319559, A330061, A330101, A330102, A330194.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
biinorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],biinorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],fbi[fbi/@#]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Table[fbi[fbi/@biinorm[bpe/@bpe[n]]],{n,0,100}]

a(n) <= n.

A371448 Numbers such that (1) the product of prime indices is squarefree, and (2) the binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 26, 30, 32, 33, 34, 40, 47, 48, 51, 52, 55, 60, 64, 66, 68, 80, 85, 86, 94, 96, 102, 104, 110, 120, 123, 127, 128, 132, 136, 141, 143, 160, 165, 170, 172, 187, 188, 192, 204, 205, 208, 215, 220, 221, 226, 240, 246

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have (1) squarefree product and (2) binary indices covering an initial interval.
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 terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}
  33: {{2},{1,3}}
  34: {{1},{1,2,3}}
  40: {{1},{1},{1},{1,2}}
  47: {{1,2,3,4}}
  48: {{1},{1},{1},{1},{2}}
  51: {{2},{1,2,3}}

An opposite version is A371293, A371292.
Without the squarefree condition we have A371447, see also A320456, A326754.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
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.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

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

Intersection of A302505 and A371447.

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

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A340018 MM-numbers of labeled graphs with half-loops covering an initial interval of positive integers, without isolated vertices.

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A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

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A330195 BII-number of the BII-normalization of the set-system with BII-number n.

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