A368112 -id:A368112

A368109

Number of ways to choose a binary index of each binary index of n.

+10
35

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, 8, 8, 8, 8, 4, 4, 4, 4, 8, 8, 8, 8, 3, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12

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

OFFSET

0,5

COMMENTS

First differs from A367912 at a(52) = 8, A367912(52) = 7.

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}.

Run-lengths are all 4 or 8.

LINKS

Table of n, a(n) for n=0..86.

FORMULA

a(n) = Product_{k in A048793(n)} A000120(k).

EXAMPLE

The binary indices of binary indices of 20 are {{1,2},{1,3}}, with choices (1,1), (1,3), (2,1), (2,3), so a(20) = 4.

The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,1,1), (1,1,3), (1,3,2), (1,3,3), (2,1,2), (2,1,3), (2,3,2), (2,3,3), so a(52) = 8.

MATHEMATICA

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

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

CROSSREFS

All entries appear to belong to A003586.

Positions of ones are A253317.

The version for prime indices is A355741, for multisets A355744.

Choosing a multiset (not sequence) gives A367912, firsts A367913.

Positions of first appearances are A368111, sorted A368112.

A048793 lists binary indices, length A000120, sum A029931.

A058891 counts set-systems, covering A003465, connected A323818.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A072639, A309326, A326031, A326702, A326753, A355731, A355739, A367771, A367905, A367906, A367915.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 12 2023

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

A367913

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

+10
10

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

(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..47.

EXAMPLE

The terms together with the corresponding set-systems begin:

1: {{1}}

4: {{1,2}}

64: {{1,2,3}}

20: {{1,2},{1,3}}

68: {{1,2},{1,2,3}}

320: {{1,2,3},{1,4}}

52: {{1,2},{1,3},{2,3}}

84: {{1,2},{1,3},{1,2,3}}

16448: {{1,2,3},{1,2,3,4}}

324: {{1,2},{1,2,3},{1,4}}

832: {{1,2,3},{1,4},{2,4}}

116: {{1,2},{1,3},{2,3},{1,2,3}}

MATHEMATICA

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

spnm[y_]:=Max@@NestWhile[Most, y, Union[#]!=Range[0, Max@@#]&];

c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]], {n, 1000}];

Table[Position[c, n][[1, 1]], {n, spnm[c]}]

CROSSREFS

A version for multisets and divisors is A355734.

With distinctness we have A367910, firsts of A367905, sorted A367911.

Positions of first appearances in A367912.

The sorted version is A367915.

For sequences we have A368111, firsts of A368109, sorted A368112.

For sets we have A368184, firsts of A368183, sorted A368185.

A048793 lists binary indices, length A000120, sum A029931.

A058891 counts set-systems, covering A003465, connected A323818.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355741, A355744.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 16 2023

STATUS

approved

A367915

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

+10
10

1, 4, 20, 52, 64, 68, 84, 116, 308, 320, 324, 340, 372, 816, 832, 836, 848, 852, 880, 884, 1104, 1108, 1136, 1360, 1364, 1392, 1396, 1904, 1908, 2868, 2884, 2900, 2932, 3152, 3184, 3188, 3412, 3424, 3440, 3444, 3952, 3956, 5188, 5204, 5216, 5220, 5236, 5476

(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..48.

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}}

320: {{1,2,3},{1,4}}

324: {{1,2},{1,2,3},{1,4}}

340: {{1,2},{1,3},{1,2,3},{1,4}}

372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}

MATHEMATICA

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

c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]], {n, 10000}];

Select[Range[Length[c]], FreeQ[Take[c, #-1], c[[#]]]&]

CROSSREFS

A version for multisets and divisors is A355734.

Sorted positions of first appearances in A367912, for sequences A368109.

The unsorted version is A367913.

A048793 lists binary indices, length A000120, sum A029931.

A058891 counts set-systems, covering A003465, connected A323818.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355744, A367905, A367906, A367911, A368112, A368185.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 16 2023

STATUS

approved

A368111

Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

+10
9

1, 4, 64, 20, 68, 52, 1088, 84, 308, 1092, 116, 5184, 820, 1108, 372, 5188, 2868, 1140, 13376, 884, 5204, 17204, 1396, 13380, 2932, 5236, 275520, 19252, 1908, 13396, 17268, 5492, 275524, 84788, 3956, 13428, 1324096, 19316, 6004, 275540, 215860, 18292, 13684

(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..43.

EXAMPLE

The terms together with the corresponding set-systems begin:

1: {{1}}

4: {{1,2}}

64: {{1,2,3}}

20: {{1,2},{1,3}}

68: {{1,2},{1,2,3}}

52: {{1,2},{1,3},{2,3}}

84: {{1,2},{1,3},{1,2,3}}

308: {{1,2},{1,3},{2,3},{1,4}}

116: {{1,2},{1,3},{2,3},{1,2,3}}

820: {{1,2},{1,3},{2,3},{1,4},{2,4}}

372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}

884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

MATHEMATICA

nn=10000;

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

dd=Select[Range[nn], Max@@First/@FactorInteger[#]<=3&];

qq=Table[Length[Tuples[bpe/@bpe[n]]], {n, nn}];

kk=Select[Range[Length[dd]], SubsetQ[qq, Take[dd, #]]&]

Table[Position[qq, dd[[n]]][[1, 1]], {n, kk}]

CROSSREFS

With distinctness we have A367910, sorted A367911, firsts of A367905.

For multisets we have A367913, sorted A367915, firsts of A367912.

Positions of first appearances in A368109.

The sorted version is A368112.

A048793 lists binary indices, length A000120, sum A029931.

A058891 counts set-systems, covering A003465, connected A323818.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A072639, A253317, A309326, A326031, A326702, A326753, A355731, A355741, A355744, A367906.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 17 2023

STATUS

approved

A368183

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

+10
8

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

(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}.

LINKS

Table of n, a(n) for n=0..86.

EXAMPLE

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

MATHEMATICA

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

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

CROSSREFS

For sequences we have A367905, firsts A367910, sorted A367911.

Positions of zeros are A367907.

Without distinctness we have A367912, firsts A367913, sorted A367915.

Positions of positive terms are A367906.

For sequences without distinctness: A368109, firsts A368111, sorted A368112.

Positions of first appearances are A368184, 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, A355739, A355741, A355744, A367771.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 17 2023

STATUS

approved

A368184

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

+10
7

7, 1, 4, 20, 276, 320, 1088, 65856, 66112, 66624, 263232

(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..10.

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}}

276: {{1,2},{1,3},{1,4}}

320: {{1,2,3},{1,4}}

1088: {{1,2,3},{1,2,4}}

65856: {{1,2,3},{1,4},{1,5}}

66112: {{1,2,3},{2,4},{1,5}}

66624: {{1,2,3},{1,2,4},{1,5}}

MATHEMATICA

nn=10000;

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

q=Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]]], {n, nn}];

k=Max@@Select[Range[Max@@q], SubsetQ[q, Range[#]]&]

Table[Position[q, n][[1, 1]], {n, 0, k}]

CROSSREFS

For strict sequences: A367910, firsts of A367905, sorted A367911.

For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.

For sequences w/o distinctness: A368111, firsts of A368109, sorted A368112.

Positions of first appearances in A368183.

The sorted version is 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, A253317, A326031, A326702, A326753, A355739, A355741, A367771, A367906, A367907.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 18 2023

STATUS

approved

A368185

Sorted list of positions of first appearances in A368183 (number of sets that can be obtained by choosing a different binary index of each binary index).

+10
6

1, 4, 7, 20, 276, 320, 1088, 65856, 66112, 66624

(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..10.

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}}

276: {{1,2},{1,3},{1,4}}

320: {{1,2,3},{1,4}}

1088: {{1,2,3},{1,2,4}}

65856: {{1,2,3},{1,4},{1,5}}

66112: {{1,2,3},{2,4},{1,5}}

66624: {{1,2,3},{1,2,4},{1,5}}

MATHEMATICA

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

c=Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]]], {n, 1000}];

Select[Range[Length[c]], FreeQ[Take[c, #-1], c[[#]]]&]

CROSSREFS

For sequences we have A367911, unsorted A367910, firsts of A367905.

Multisets w/o distinctness: A367915, unsorted A367913, firsts of A367912.

Sequences w/o distinctness: A368112, unsorted A368111, firsts of A368109.

Sorted list of positions of first appearances in A368183.

The unsorted version is 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, A253317, A326031, A326702, A326753, A355739, A355741, A367771, A367906, A367907.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 18 2023

STATUS

approved