A368109 -id:A368109

Search: a368109 -id:a368109

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A368112

Sorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).

+20
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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}.

LINKS

Table of n, a(n) for n=1..44.

EXAMPLE

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}}

MATHEMATICA

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[[#]]]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 17 2023

STATUS

approved

A370636

Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

+10
27

1, 2, 4, 7, 14, 24, 39, 61, 122, 203, 315, 469, 676, 952, 1307, 1771, 3542, 5708, 8432, 11877, 16123, 21415, 27835, 35757, 45343, 57010, 70778, 87384, 106479, 129304, 155802, 187223, 374446, 588130, 835800, 1124981, 1456282, 1841361, 2281772, 2791896, 3367162

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..40.

Wikipedia, Axiom of choice.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

FORMULA

a(2^n - 1) = A367902(n).

Partial sums of A370639.

EXAMPLE

The a(0) = 1 through a(4) = 14 subsets:

{} {} {} {} {}

{1} {1} {1} {1}

{2} {2} {2}

{1,2} {3} {3}

{1,2} {4}

{1,3} {1,2}

{2,3} {1,3}

{1,4}

{2,3}

{2,4}

{3,4}

{1,2,4}

{1,3,4}

{2,3,4}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#], UnsameQ@@#&]!={}&]], {n, 0, 10}]

CROSSREFS

Simple graphs of this type are counted by A133686, covering A367869.

Unlabeled graphs of this type are counted by A134964, complement A140637.

Simple graphs not of this type are counted by A367867, covering A367868.

Set systems of this type are counted by A367902, ranks A367906.

Set systems not of this type are counted by A367903, ranks A367907.

Set systems uniquely of this type are counted by A367904, ranks A367908.

Unlabeled multiset partitions of this type are A368098, complement A368097.

A version for MM-numbers of multisets is A368100, complement A355529.

Factorizations are counted by A368414/A370814, complement A368413/A370813.

For prime indices we have A370582, differences A370586.

The complement for prime indices is A370583, differences A370587.

The complement is A370637, differences A370589, without ones A370643.

The case of a unique choice is A370638, maxima A370640, differences A370641.

First differences are A370639.

The minimal case of the complement 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, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368095, A368109.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 08 2024

EXTENSIONS

a(19)-a(40) from Alois P. Heinz, Mar 09 2024

STATUS

approved

A367912

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

+10
22

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..87.

EXAMPLE

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.

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]], {n, 0, 100}]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 12 2023

STATUS

approved

A370637

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

+10
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..34.

Wikipedia, Axiom of choice.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

FORMULA

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

Partial sums of A370589.

EXAMPLE

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}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#], UnsameQ@@#&]=={}&]], {n, 0, 10}]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 08 2024

EXTENSIONS

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

STATUS

approved

A253317

Indices in A261283 where records occur.

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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: 0 ~ {}

1: 1 ~ {1}

2: 10 ~ {2}

3: 11 ~ {1,2}

8: 1000 ~ {4}

9: 1001 ~ {1,4}

10: 1010 ~ {2,4}

11: 1011 ~ {1,2,4}

128: 10000000 ~ {8}

129: 10000001 ~ {1,8}

130: 10000010 ~ {2,8}

131: 10000011 ~ {1,2,8}

136: 10001000 ~ {4,8}

137: 10001001 ~ {1,4,8}

138: 10001010 ~ {2,4,8}

139: 10001011 ~ {1,2,4,8}

For powers of 3 we have A368531.

(End)

LINKS

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.

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(PARI) a(n)={if(n<=1, 0, my(t=1<<logint(n-1, 2)); a(n-t) + 2^(t-1))} \\ Andrew Howroyd, Dec 20 2019

CROSSREFS

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.

KEYWORD

nonn,base

AUTHOR

Philippe Beaudoin, Dec 30 2014

EXTENSIONS

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

STATUS

approved

A370638

Number of subsets of {1..n} such that a unique set can be obtained by choosing a different binary index of each element.

+10
15

1, 2, 4, 6, 12, 19, 30, 45, 90, 147, 230, 343, 504, 716, 994, 1352, 2704, 4349, 6469

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..18.

Wikipedia, Axiom of choice.

FORMULA

a(2^n - 1) = A370818(n).

EXAMPLE

The set {3,4} has binary indices {{1,2},{3}}, with two choices {1,3}, {2,3}, so is not counted under a(4).

The a(0) = 1 through a(5) = 19 subsets:

{} {} {} {} {} {}

{1} {1} {1} {1} {1}

{2} {2} {2} {2}

{1,2} {1,2} {4} {4}

{1,3} {1,2} {1,2}

{2,3} {1,3} {1,3}

{1,4} {1,4}

{2,3} {1,5}

{2,4} {2,3}

{1,2,4} {2,4}

{1,3,4} {4,5}

{2,3,4} {1,2,4}

{1,2,5}

{1,3,4}

{1,3,5}

{2,3,4}

{2,3,5}

{2,4,5}

{3,4,5}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Table[Length[Select[Subsets[Range[n]], Length[Union[Sort /@ Select[Tuples[bpe/@#], UnsameQ@@#&]]]==1&]], {n, 0, 10}]

CROSSREFS

Set systems of this type are counted by A367904, ranks A367908.

A version for MM-numbers of multisets is A368101.

For prime indices we have A370584.

This is the unique version of A370636, complement A370637.

The maximal case is A370640, differences A370641.

Factorizations of this type are counted by A370645.

The case A370818 is the restriction to A000225.

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.

Cf. A133686, A134964, A326031, A326702, A367772, A367867, A367905, A367909, A367912, A368109.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Mar 09 2024

STATUS

approved

A371452

Number of connected components of the prime indices of the binary indices of n.

+10
14

1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

The prime indices of binary indices of 281492156579880 are {{1,1},{1,2},{3,4},{4,4}}, with 2 connected components {{1,1},{1,2}} and {{3,4},{4,4}}, so a(281492156579880) = 2.

MATHEMATICA

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Length[csm[prix/@bix[n]]], {n, 100}]

CROSSREFS

Positions of first appearances are A080355, opposite A325782.

For prime indices of prime indices we have A305079, ones A305078.

For binary indices of binary indices we have A326753, ones A326749.

Positions of ones are A371291.

For binary indices of prime indices we have A371451, ones A325118.

A001187 counts connected graphs.

A007718 counts non-isomorphic connected multiset partitions.

A048143 counts connected antichains of sets.

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.

A326964 counts connected set-systems, covering A323818.

Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 01 2024

STATUS

approved

A367911

Sorted positions of first appearances in A367905.

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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}.

LINKS

Table of n, a(n) for n=1..18.

EXAMPLE

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}}

MATHEMATICA

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[[#]]]&]

CROSSREFS

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.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 16 2023

STATUS

approved

A367910

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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}.

LINKS

Table of n, a(n) for n=0..14.

Wikipedia, Axiom of choice.

EXAMPLE

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}}

MATHEMATICA

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]}]

CROSSREFS

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.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 16 2023

STATUS

approved

A370640

Number of maximal subsets of {1..n} such that it is possible to choose a different binary index of each element.

+10
12

1, 1, 1, 3, 3, 8, 17, 32, 32, 77, 144, 242, 383, 580, 843, 1201, 1201, 2694, 4614, 7096, 10219, 14186, 19070, 25207, 32791, 42160

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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.

Also choices of A029837(n) elements of {1..n} such that it is possible to choose a different binary index of each.

LINKS

Table of n, a(n) for n=0..25.

EXAMPLE

The a(0) = 1 through a(6) = 17 subsets:

{} {1} {1,2} {1,2} {1,2,4} {1,2,4} {1,2,4}

{1,3} {1,3,4} {1,2,5} {1,2,5}

{2,3} {2,3,4} {1,3,4} {1,2,6}

{1,3,5} {1,3,4}

{2,3,4} {1,3,5}

{2,3,5} {1,3,6}

{2,4,5} {1,4,6}

{3,4,5} {1,5,6}

{2,3,4}

{2,3,5}

{2,3,6}

{2,4,5}

{2,5,6}

{3,4,5}

{3,4,6}

{3,5,6}

{4,5,6}

The a(0) = 1 through a(6) = 17 set-systems:

{1} {1}{2} {1}{2} {1}{2}{3} {1}{2}{3} {1}{2}{3}

{1}{12} {1}{12}{3} {1}{12}{3} {1}{12}{3}

{2}{12} {2}{12}{3} {1}{2}{13} {1}{2}{13}

{2}{12}{3} {1}{2}{23}

{2}{3}{13} {1}{3}{23}

{1}{12}{13} {2}{12}{3}

{12}{3}{13} {2}{3}{13}

{2}{12}{13} {1}{12}{13}

{1}{12}{23}

{1}{13}{23}

{12}{3}{13}

{12}{3}{23}

{2}{12}{13}

{2}{12}{23}

{2}{13}{23}

{3}{13}{23}

{12}{13}{23}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Table[Length[Select[Subsets[Range[n], {IntegerLength[n, 2]}], Select[Tuples[bpe/@#], UnsameQ@@#&]!={}&]], {n, 0, 10}]

CROSSREFS

Dominated by A357812.

The version for set-systems is A368601, max of A367902 (complement A367903).

For prime indices we have A370585, with n A370590, see also A370591.

This is the maximal case of A370636 (complement A370637).

The case of a unique choice is A370638.

The case containing n is A370641, non-maximal A370639.

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.

A307984 counts Q-bases of logarithms of positive integers.

A355741 counts choices of a prime factor of each prime index.

Cf. A133686, A326031, A326702, A367905, A367909, A367912, A368109, A368110, A370592, A370642.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Mar 10 2024

STATUS

approved

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