A072639 -id:A072639 - cp's OEIS frontend

A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 4, 4, 4, 4, 7, 7, 7, 7, 3, 3, 3, 3, 5, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 8, 8, 8, 8

Offset: 0

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. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

The run-lengths are all 4 or 8.

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with multiset choices {1,1,2}, {1,1,3}, {1,2,2}, {1,2,3}, {1,3,3}, {2,2,3}, {2,3,3}, so a(52) = 7.

Positions of ones are A253317.
The version for multisets and divisors is A355733, for sequences A355731.
The version for multisets is A355744, for sequences A355741.
For a sequence of distinct choices we have A367905, firsts A367910.
Positions of first appearances are A367913, sorted A367915.
Choosing a sequence instead of multiset gives A368109, firsts A368111.
Choosing a set instead of multiset gives A368183, firsts A368184.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355735, A355739, A355740, A355745, A367771, A367906.

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

A327041 a(n) is the number whose binary indices are the union of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 3, 3, 4, 5, 6, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Gus Wiseman, Aug 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 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.

			22 is the BII-number of {{2},{1,2},{1,3}}, and 7 has binary indices {1,2,3}, so a(22) = 7.

Tilman Piesk, Illustration of the first 128 terms

Indices of records are A253317.

Cf. A000120, A001511, A029931, A035327, A048793, A070939, A072639, A326031, A326702, A326947, A261283 (XOR equivalent).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[2^Union@@bpe/@bpe[n]]/2,{n,0,100}]

A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 25, 67, 134, 309, 709, 1579, 3420, 7240, 15077, 30997, 61994, 125364, 253712, 512411, 1032453, 2075737, 4166469, 8352851, 16731873, 33497422, 67038086, 134130344, 268328977, 536741608, 1073586022, 2147296425, 4294592850, 8589346462, 17179033384

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(0) = 0 through a(5) = 8 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                    {1,2,3,4}  {1,4,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}
                               {1,2,3,4,5}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs not of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A140637, complement A134964.
Simple graphs of this type are counted by A367867, covering A367868.
Set systems not of this type are counted by A367902, ranks A367906.
Set systems of this type are counted by A367903, ranks A367907.
Set systems uniquely not of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368097, complement A368098.
A version for MM-numbers of multisets is A355529, complement A368100.
Factorizations are counted by A368413/A370813, complement A368414/A370814.
The complement for prime indices is A370582, differences A370586.
For prime indices we have A370583, differences A370587.
First differences are A370589.
The complement is counted by A370636, differences A370639.
The case without ones is A370643.
The version for a unique choice is A370638, maxima A370640, diffs A370641.
The minimal case is A370642, without ones A370644.
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, A072639, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

a(2^n - 1) = A367903(n).

Partial sums of A370589.

a(21)-a(34) from Alois P. Heinz, Mar 09 2024

A072638 Number of unary-binary rooted trees of height at most n.

Original entry on oeis.org

0, 1, 3, 10, 66, 2278, 2598060, 3374961778891, 5695183504492614029263278, 16217557574922386301420536972254869595782763547560

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

A unary-binary tree is one in which the degree of every node is <= 3.
a(n+1) = (a(n)+1)-th triangular numbers = A000217(a(n)+1). a(n+1) = (a(n) + 1) * (a(n) + 2) / 2. a(n+1) = A006894(n+2) - 1. - Jaroslav Krizek, Sep 11 2009
a(n) is the smallest integer that is the sum of n distinct members of the complete sequence A000124. See A204009 for the binary vectors that select the terms from A000124. - Frank M Jackson, Jan 09 2012

Index entries for sequences related to rooted trees

Maximal position in A071673 where the value n occurs.

Binary width of each term: A072641. Cf. A072639, A072640, A072654.

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=binomial(a[n-1]+2,2) od: seq(a[n], n=-1..9); # Zerinvary Lajos, Jun 08 2007

Clear[a]; a[0] = 0; a[n_] := a[n] = 1 + (a[n-1]*(a[n-1]+3))/2; Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Jan 31 2013 *)

a(n+1) = 1 + (a(n)*(a(n)+3))/2.
Conjecture: a(n) = A006894(n+1) - 1. - R. J. Mathar, Apr 23 2007
a(n) := C(a(n-1) + 2, 2), n >= -1. - Zerinvary Lajos, Jun 08 2007

Edited by Christian G. Bower, Oct 23 2002

A253317 Indices in A261283 where records occur.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 10, 11, 128, 129, 130, 131, 136, 137, 138, 139, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906, 32907, 2147483648, 2147483649, 2147483650, 2147483651, 2147483656, 2147483657

Offset: 1

Philippe Beaudoin, Dec 30 2014

From Gus Wiseman, Dec 29 2023: (Start)
These are numbers whose binary indices are all powers of 2, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, the terms together with their binary expansions and binary indices begin:
       0 ~ {}
       1 ~ {1}
      10 ~ {2}
      11 ~ {1,2}
    1000 ~ {4}
    1001 ~ {1,4}
    1010 ~ {2,4}
    1011 ~ {1,2,4}
10000000 ~ {8}
10000001 ~ {1,8}
10000010 ~ {2,8}
10000011 ~ {1,2,8}
10001000 ~ {4,8}
10001001 ~ {1,4,8}
10001010 ~ {2,4,8}
10001011 ~ {1,2,4,8}
For powers of 3 we have A368531.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..4096
Lorenzo Sauras-Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.

Cf. A053644 (most significant bit).
A048793 lists binary indices, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A058891, A062050, A072639, A326031, A326675, A326702, A367771, A367912, A368183, A368109, A368531.

a := proc(n) local k, A:
A := [seq(0,i=1..n)]: A[1]:=0:
for k from 1 to n-1 do
   A[k+1] := A[k-2^ilog2(k)+1]+2^(2^ilog2(k)-1): od:
return A[n]: end proc: # Lorenzo Sauras Altuzarra, Dec 18 2019
# second Maple program:
a:= n-> (l-> add(l[i+1]*2^(2^i-1), i=0..nops(l)-1))(Bits[Split](n-1)):
seq(a(n), n=1..38);  # Alois P. Heinz, Dec 13 2023

Nest[Append[#1, #1[[-#2]] + 2^(#2 - 1)] & @@ {#, 2^(IntegerLength[Length[#], 2] - 1)} &, {0, 1}, 36] (* Michael De Vlieger, May 08 2020 *)

a(n)={if(n<=1, 0, my(t=1<Andrew Howroyd, Dec 20 2019

a(1) = 0 and a(n) = a(n-A053644(n-1)) + 2^(A053644(n-1)-1). - Lorenzo Sauras Altuzarra, Dec 18 2019
a(n) = A358126(n-1) / 2. - Tilman Piesk, Dec 18 2022
a(2^n+1) = 2^(2^n-1) = A058891(n+1). - Gus Wiseman, Dec 29 2023
a(2^n) = A072639(n). - Gus Wiseman, Dec 29 2023
G.f.: 1/(1-x) * Sum_{k>=0} (2^(-1+2^k))*x^2^k/(1+x^2^k). - John Tyler Rascoe, May 22 2024

Corrected reference in name from A253315 to A261283. - Tilman Piesk, Dec 18 2022

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.

A367911 Sorted positions of first appearances in A367905.

Original entry on oeis.org

1, 4, 7, 20, 68, 320, 352, 1088, 3136, 5184, 13376, 16704, 17472, 70720, 82240, 83008, 90112, 90176

Offset: 1

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:
      1: {{1}}
      4: {{1,2}}
      7: {{1},{2},{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}}
   5184: {{1,2,3},{1,2,4},{1,3,4}}
  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  16704: {{1,2,3},{1,4},{1,2,3,4}}
  17472: {{1,2,3},{1,2,4},{1,2,3,4}}
  70720: {{1,2,3},{1,2,4},{1,3,4},{1,5}}
  82240: {{1,2,3},{1,4},{1,2,3,4},{1,5}}

Sorted positions of first appearances in A367905.
The unsorted version is A367910.
Multisets without distinctness are A367915, unsorted A367913.
Without distinctness we have A368112, unsorted A368111.
For sets instead of sequences we have A368185, unsorted A368184.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A367902, A367906, A367907, A367912, A368109, A368183.

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

A072644 Size of the parenthesizations obtained with the global ranking/unranking scheme A072634-A072637.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 4, 3, 4, 3, 4, 5, 5, 5, 5, 4, 4, 5, 5, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 6, 7, 4, 5, 4, 5, 6, 6, 6, 6, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 7, 8, 7, 7, 8, 8, 7, 8, 8, 9, 5, 4, 6, 5, 5, 4, 6, 5, 7, 6, 7, 6, 7, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 8, 8, 8, 7, 8, 7, 8, 7, 8, 7

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

Sohrab Towfighi, Symbolic regression by random search, arXiv:1906.07848 [cs.NE], 2019.

Cf. A072635 & A072637. A072644(n) = A029837(A014486(A072635(n))+1)/2 or = A029837(A014486(A072637(n))+1)/2 [A029837(n+1) gives the binary width of n].

Each value v occurs A000108(v) times. The maximum position for value v to occur is A072639(v). Permutations: A071673, A072643, A072645, A072660.

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]}]

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[#]]&]

cp's OEIS Frontend

A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

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A327041 a(n) is the number whose binary indices are the union of the set-system with BII-number n.

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A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

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A072638 Number of unary-binary rooted trees of height at most n.

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A253317 Indices in A261283 where records occur.

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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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A367911 Sorted positions of first appearances in A367905.

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A072644 Size of the parenthesizations obtained with the global ranking/unranking scheme A072634-A072637.

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