A327041 -id:A327041 - cp's OEIS frontend

A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 7, 14, 21, 42, 28, 56, 84, 168, 35, 70, 105, 210, 140, 280, 420, 840, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 63, 126, 189, 378, 252, 504, 756, 1512, 315, 630, 945, 1890

Offset: 0

Christian G. Bower, Dec 15 1999

Inverse of sequence A064358 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.
Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005
4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005
Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011
The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018
Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k). 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. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019
The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019

			Terms following 5 are 10, 15, 30, 20, 40, 60, 120; this is followed by 7 as 6 has already occurred. - _Philippe Deléham_, Jun 03 2015
From _Antti Karttunen_, Apr 13 2018, after also _Philippe Deléham_'s Jun 03 2015 example: (Start)
This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A300841(k), and each right hand child contains 2*A300841(k), when their parent contains k:
                                     1
                                     |
                  ...................2...................
                 3                                       6
       4......../ \........8                  12......../ \........24
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   5       10         15       30         20       40         60      120
  7 14   21  42     28  56   84  168    35  70  105  210   140 280  420 840
  etc.
Compare also to trees like A005940 and A283477, and sequences A207901 and A302783.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191 (Terms 0..1023 from T. D. Noe)
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..16383, with a color function representing primes in red, perfect powers of primes in gold, squarefree composites in greens, and numbers neither squarefree nor prime powers in blue or purple. Purple represents powerful numbers that are not prime powers. This is a 15 level version of Karttunen's diagram in the example.
OEIS Wiki, Ordering of positive integers by increasing "Fermi-Dirac representation"
Index entries for sequences that are permutations of the natural numbers

Subsequences: A019565 (squarefree terms), A050376 (the left edge from 2 onward), A336882 (odd terms).
Cf. A050030, A052331 (inverse), A096111, A096113, A096114, A096115, A096116, A096118, A096119, A182979, A207901, A300841, A302023, A302783.
Cf. A000120, A029931, A048793, A064547, A070939, A213925, A299755, A299757, A327041.

a = {1}; Do[a = Join[a, a*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)

up_to_e = 13; \\ Good for computing up to n = (2^13)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); }; \\ Antti Karttunen, Apr 12 2018

a(0)=1; a(n+2^k)=a(n)*b(k) for n < 2^k, k = 0, 1, ... where b is A050376. - Thomas Ordowski, Mar 04 2005
The binary representation of n, n = Sum_{i=0..1+floor(log_2(n))} n_i * 2^i, n_i in {0,1}, is taken as the "Fermi-Dirac representation" (A182979) of a(n), a(n) = Product_{i=0..1+floor(log_2(n))} (b_i)^(n_i) where b_i is A050376(i), i.e., the i-th "Fermi-Dirac prime" (prime power with exponent being a power of 2). - Daniel Forgues, Feb 12 2011
From Antti Karttunen, Apr 12 & 17 2018: (Start)
a(0) = 1; a(2n) = A300841(a(n)), a(2n+1) = 2*A300841(a(n)).
a(n) = A207901(A006068(n)) = A302783(A003188(n)) = A302781(A302845(n)).
(End)

Entry revised Mar 17 2005 by N. J. A. Sloane, based on comments from several people, especially David Wasserman and Thomas Ordowski

A327111 BII-numbers of set-systems with spanning edge-connectivity 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 88, 89, 90, 91, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Aug 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 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.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-connectivity 1 together with their BII-numbers begins:
   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}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  28: {{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}}
  32: {{2,3}}

Graphs with spanning edge-connectivity >= 2 are counted by A095983.
BII-numbers for vertex-connectivity 1 are A327098.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
Graphs with spanning edge-connectivity 1 are counted by A327145.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Cf. A013922, A322395, A326749, A327041, A327069, A327071, A327097, A327144, A327145.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]

A327144 Spanning edge-connectivity of the set-system with BII-number n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 31 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.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Positions of first appearances of each integer together with the corresponding set-systems:
     0: {}
     1: {{1}}
    52: {{1,2},{1,3},{2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
  3952: {{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4}}
  8052: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4}}

Dominated by A327103.
The same for cut-connectivity is A326786.
The same for non-spanning edge-connectivity is A326787.
The same for vertex-connectivity is A327051.
Positions of 1's are A327111.
Positions of 2's are A327108.
Positions of first appearance of each integer are A327147.
Cf. A000120, A048793, A070939, A322338, A323818, A326031, A327041, A327069, A327076, A327130, A327145.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[spanEdgeConn[Union@@bpe/@bpe[n],bpe/@bpe[n]],{n,0,100}]

A326853 BII-numbers of set-systems where every two covered vertices appear together in some edge (cointersecting).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 24, 25, 32, 34, 40, 42, 52, 53, 54, 55, 60, 61, 62, 63, 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

Offset: 1

Gus Wiseman, Aug 18 2019

nonn

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence gives all BII-numbers (defined below) of set-systems that are cointersecting, meaning their dual is pairwise intersecting.

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.

			The sequence of all cointersecting set-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}}
  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}}

BII-numbers of pairwise intersecting set-systems are A326910.
Cointersecting set-systems are A327039, with covering version A327040.
The T_0 or costrict case is A327052.
Cf. A029931, A048793, A326031, A327020, A327037, A327041, A327053, A327057.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
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],stableQ[dual[bpe/@bpe[#]],Intersection[#1,#2]=={}&]&]

A327108 BII-numbers of set-systems with spanning edge-connectivity 2.

Original entry on oeis.org

52, 53, 54, 55, 60, 61, 62, 63, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113, 114, 115, 120, 121, 122, 123, 772, 773, 774, 775, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 848, 849, 850

Offset: 1

Gus Wiseman, Aug 23 2019

nonn

Differs from A327109 in lacking 116, 117, 118, 119, 124, 125, 126, 127, ...
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.
The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-connectivity 2 together with their BII-numbers begins:
   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}}
   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}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   86: {{2},{1,2},{1,3},{1,2,3}}
   87: {{1},{2},{1,2},{1,3},{1,2,3}}
   92: {{1,2},{3},{1,3},{1,2,3}}
   93: {{1},{1,2},{3},{1,3},{1,2,3}}
   94: {{2},{1,2},{3},{1,3},{1,2,3}}
   95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  101: {{1},{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}
  103: {{1},{2},{1,2},{2,3},{1,2,3}}

Positions of 2's in A327144.
Graphs with spanning edge-connectivity >= 2 are counted by A095983.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A326749, A326787, A327041, A327069, A327075, A327102, A327103, A327104.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==2&]

A327103 Minimum vertex-degree in the set-system with BII-number n.

Original entry on oeis.org

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

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) = 2.

The maximum vertex-degree is A327104.
Positions of 1's are A327105.
Positions of terms > 1 are A327107.
Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326783, A326786, A327041, A327228, A327229.

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

A327097 BII-numbers of set-systems with non-spanning edge-connectivity 2.

Original entry on oeis.org

5, 6, 17, 20, 24, 34, 36, 40, 48, 53, 54, 55, 60, 61, 62, 63, 65, 66, 68, 71, 72, 80, 86, 87, 89, 92, 93, 94, 95, 96, 101, 103, 106, 108, 109, 110, 111, 113, 114, 115, 121, 122, 123, 257, 260, 272, 308, 309, 310, 311, 316, 317, 318, 319, 320, 326, 327, 342

Offset: 1

Gus Wiseman, Aug 20 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.

The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any isolated vertices) to result in a disconnected or empty set-system.

			The sequence of all set-systems with non-spanning edge-connectivity 2 together with their BII-numbers begins:
   5: {{1},{1,2}}
   6: {{2},{1,2}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  24: {{3},{1,3}}
  34: {{2},{2,3}}
  36: {{1,2},{2,3}}
  40: {{3},{2,3}}
  48: {{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}}
  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}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}

Positions of 2's in A326787.
BII-numbers for vertex-connectivity 2 are A327082.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for non-spanning edge-connectivity > 1 are A327102.
BII-numbers for spanning edge-connectivity 2 are A327108.
Cf. A007146, A048793, A052446, A059166, A070939, A095983, A263296, A322335, A322338, A322395, A326031, A327041, A327069, A327111.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
Select[Range[0,100],edgeConn[bpe[#]]==2&]

A327109 BII-numbers of set-systems with spanning edge-connectivity >= 2.

Original entry on oeis.org

52, 53, 54, 55, 60, 61, 62, 63, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 772, 773, 774, 775, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826

Offset: 1

Gus Wiseman, Aug 23 2019

nonn

Differs from A327108 in having 116, 117, 118, 119, 124, 125, 126, 127, ...
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.
The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-connectivity >= 2 together with their BII-numbers begins:
   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}}
   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}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   86: {{2},{1,2},{1,3},{1,2,3}}
   87: {{1},{2},{1,2},{1,3},{1,2,3}}
   92: {{1,2},{3},{1,3},{1,2,3}}
   93: {{1},{1,2},{3},{1,3},{1,2,3}}
   94: {{2},{1,2},{3},{1,3},{1,2,3}}
   95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  101: {{1},{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}
  103: {{1},{2},{1,2},{2,3},{1,2,3}}

Positions of terms >= 2 in  A327144.
Graphs with spanning edge-connectivity >= 2 are counted by A095983.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for non-spanning edge-connectivity >= 2 are A327102.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A326749, A326753, A326787, A327041, A327069, A327071, A327075.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,1000],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]>=2&]

A327082 BII-numbers of set-systems with cut-connectivity 2.

Original entry on oeis.org

4, 5, 6, 7, 16, 17, 24, 25, 32, 34, 40, 42, 256, 257, 384, 385, 512, 514, 640, 642, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850

Offset: 1

Gus Wiseman, Aug 20 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

We define the cut-connectivity (A326786, A327237), of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex and no edges has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039), this is the same as vertex-connectivity (A327334, A327051).

			The sequence of all set-systems with cut-connectivity 2 together with their BII-numbers begins:
    4: {{1,2}}
    5: {{1},{1,2}}
    6: {{2},{1,2}}
    7: {{1},{2},{1,2}}
   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}}
  256: {{1,4}}
  257: {{1},{1,4}}
  384: {{4},{1,4}}
  385: {{1},{4},{1,4}}
  512: {{2,4}}
  514: {{2},{2,4}}
  640: {{4},{2,4}}
  642: {{2},{4},{2,4}}
The first term involving an edge of size 3 is 832: {{1,2,3},{1,4},{2,4}}.

Positions of 2's in A326786.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity 2 are A327108.
The cut-connectivity 1 version is A327098.
The cut-connectivity > 1 version is A327101.
Covering 2-cut-connected set-systems are counted by A327112.
Covering set-systems with cut-connectivity 2 are counted by A327113.
Cf. A000120, A002218, A013922, A048793, A259862, A322387, A322388, A326031, A327041, A327114.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Select[Range[0,100],vertConnSys[bpe/@bpe[#]]==2&]

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&]

cp's OEIS Frontend

A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.

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A327111 BII-numbers of set-systems with spanning edge-connectivity 1.

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A327144 Spanning edge-connectivity of the set-system with BII-number n.

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A326853 BII-numbers of set-systems where every two covered vertices appear together in some edge (cointersecting).

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A327108 BII-numbers of set-systems with spanning edge-connectivity 2.

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A327103 Minimum vertex-degree in the set-system with BII-number n.

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A327097 BII-numbers of set-systems with non-spanning edge-connectivity 2.

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A327109 BII-numbers of set-systems with spanning edge-connectivity >= 2.

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A327082 BII-numbers of set-systems with cut-connectivity 2.

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