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Search: a370589 -id:a370589
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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
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
A370587
Number of subsets of {1..n} containing n such that it is not possible to choose a different prime factor of each element (non-choosable).
+10
13
0, 1, 1, 2, 6, 10, 24, 44, 116, 236, 468, 908, 1960, 3776, 7812, 15876, 32504, 63744, 130104
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..18.
EXAMPLE
The a(0) = 0 through a(5) = 10 subsets:
. {1} {1,2} {1,3} {1,4} {1,5}
{1,2,3} {2,4} {1,2,5}
{1,2,4} {1,3,5}
{1,3,4} {1,4,5}
{2,3,4} {2,4,5}
{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
Table[Length[Select[Subsets[Range[n]], MemberQ[#, n] && Length[Select[Tuples[If[#==1, {}, First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==0&]], {n, 0, 10}]
CROSSREFS
First differences of A370583, complement A370582, cf. A370584.
The complement is counted by A370586.
For a unique choice we have A370588.
For binary indices instead of factors we have A370639, complement A370589.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370585 counts maximal choosable sets.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367905, A370636.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 28 2024
STATUS
approved
A370639
Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.
+10
11
0, 1, 2, 3, 7, 10, 15, 22, 61, 81, 112, 154, 207, 276, 355, 464, 1771, 2166, 2724, 3445, 4246, 5292, 6420, 7922, 9586, 11667, 13768, 16606, 19095, 22825, 26498, 31421, 187223, 213684, 247670, 289181, 331301, 385079, 440411, 510124, 575266, 662625, 747521
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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..42.
Wikipedia, Axiom of choice.
FORMULA
First differences of A370636.
EXAMPLE
The a(0) = 0 through a(6) = 15 subsets:
. {1} {2} {3} {4} {5} {6}
{1,2} {1,3} {1,4} {1,5} {1,6}
{2,3} {2,4} {2,5} {2,6}
{3,4} {3,5} {3,6}
{1,2,4} {4,5} {4,6}
{1,3,4} {1,2,5} {5,6}
{2,3,4} {1,3,5} {1,2,6}
{2,3,5} {1,3,6}
{2,4,5} {1,4,6}
{3,4,5} {1,5,6}
{2,3,6}
{2,5,6}
{3,4,6}
{3,5,6}
{4,5,6}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Select[Subsets[Range[n]], MemberQ[#, 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 of this type are A368414/A370814, complement A368413/A370813.
For prime instead of binary indices we have A370586, differences of A370582.
The complement for prime indices is A370587, differences of A370583.
The complement is counted by A370589, differences of A370637.
Partial sums are A370636.
The complement has partial sums A370637/A370643, minima A370642/A370644.
The case of a unique choice is A370641, differences of A370638.
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, A367770, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 08 2024
EXTENSIONS
a(19)-a(42) from Alois P. Heinz, Mar 09 2024
STATUS
approved
A370642
Number of minimal subsets of {1..n} such that it is not possible to choose a different binary index of each element.
+10
9
0, 0, 0, 1, 1, 3, 9, 26, 26, 40, 82, 175, 338, 636, 1114
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
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..14.
EXAMPLE
The a(0) = 0 through a(6) = 9 subsets:
. . . {1,2,3} {1,2,3} {1,2,3} {1,2,3}
{1,4,5} {1,4,5}
{2,3,4,5} {2,4,6}
{1,2,5,6}
{1,3,4,6}
{1,3,5,6}
{2,3,4,5}
{2,3,5,6}
{3,4,5,6}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
fasmin[y_]:=Complement[y, Union@@Table[Union[s, #]& /@ Rest[Subsets[Complement[Union@@y, s]]], {s, y}]];
Table[Length[fasmin[Select[Subsets[Range[n]], Select[Tuples[bpe/@#], UnsameQ@@#&]=={}&]]], {n, 0, 10}]
CROSSREFS
For prime indices we have A370591, minima of A370583, complement A370582.
This is the minimal case of A370637, complement A370636.
The version for a unique choice is A370638, maxima A370640, diffs A370641.
The case without ones is A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368100 ranks choosable multisets, complement A355529.
A370585 counts maximal choosable sets.
Cf. A072639, A140637, A367905, A368109, A370589, A370593, A370639.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 10 2024
STATUS
approved
A370641
Number of maximal subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.
+10
8
0, 1, 1, 2, 3, 5, 9, 15, 32, 45, 67, 98, 141, 197, 263, 358, 1201, 1493, 1920, 2482, 3123, 3967, 4884, 6137, 7584, 9369
(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} containing 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) = 0 through a(7) = 15 subsets:
. {1} {1,2} {1,3} {1,2,4} {1,2,5} {1,2,6} {1,2,7}
{2,3} {1,3,4} {1,3,5} {1,3,6} {1,3,7}
{2,3,4} {2,3,5} {1,4,6} {1,4,7}
{2,4,5} {1,5,6} {1,5,7}
{3,4,5} {2,3,6} {1,6,7}
{2,5,6} {2,3,7}
{3,4,6} {2,4,7}
{3,5,6} {2,5,7}
{4,5,6} {2,6,7}
{3,4,7}
{3,5,7}
{3,6,7}
{4,5,7}
{4,6,7}
{5,6,7}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Select[Subsets[Range[n], {IntegerLength[n, 2]}], MemberQ[#, n] && Length[Union[Sort/@Select[Tuples[bpe/@#], UnsameQ@@#&]]]>0&]], {n, 0, 25}]
CROSSREFS
A version for set-systems is A368601.
For prime indices we have A370590, without n A370585, see also A370591.
This is the maximal case of A370636 requiring n, complement A370637.
This is the maximal case of A370639, complement A370589.
Without requiring n we have A370640.
Dominated by A370819.
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.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
Cf. A133686, A326031, A326702, A357812, A367905, A368100, A368109, A370586, A370638, A370642.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 11 2024
STATUS
approved
A370591
Number of minimal subsets of {1..n} such that it is not possible to choose a different prime factor of each element (non-choosable).
+10
7
0, 1, 1, 1, 2, 2, 4, 4, 7, 11, 16, 16, 30, 30, 39, 73
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..15.
EXAMPLE
The a(1) = 1 through a(10) = 16 subsets:
{1} {1} {1} {1} {1} {1} {1} {1} {1} {1}
{2,4} {2,4} {2,4} {2,4} {2,4} {2,4} {2,4}
{2,3,6} {2,3,6} {2,8} {2,8} {2,8}
{3,4,6} {3,4,6} {4,8} {3,9} {3,9}
{2,3,6} {4,8} {4,8}
{3,4,6} {2,3,6} {2,3,6}
{3,6,8} {2,6,9} {2,6,9}
{3,4,6} {3,4,6}
{3,6,8} {3,6,8}
{4,6,9} {4,6,9}
{6,8,9} {6,8,9}
{2,5,10}
{4,5,10}
{5,8,10}
{3,5,6,10}
{5,6,9,10}
MATHEMATICA
Table[Length[fasmin[Select[Subsets[Range[n]], Length[Select[Tuples[prix/@#], UnsameQ@@#&]]==0&]]], {n, 0, 15}]
CROSSREFS
Minimal case of A370583, complement A370582.
For binary indices instead of factors we have A370642, minima of A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370585 counts maximal choosable sets.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A045778, A133686, A355739, A355744, A355745, A367771, A370584, A370586, A370587, A370589.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 28 2024
STATUS
approved
A370644
Number of minimal subsets of {2..n} such that it is not possible to choose a different binary index of each element.
+10
6
0, 0, 0, 0, 0, 1, 4, 13, 13, 26, 56, 126, 243, 471, 812, 1438
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
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..15.
EXAMPLE
The a(0) = 0 through a(7) = 13 subsets:
. . . . . {2,3,4,5} {2,4,6} {2,4,6}
{2,3,4,5} {2,3,4,5}
{2,3,5,6} {2,3,4,7}
{3,4,5,6} {2,3,5,6}
{2,3,5,7}
{2,3,6,7}
{2,4,5,7}
{2,5,6,7}
{3,4,5,6}
{3,4,5,7}
{3,4,6,7}
{3,5,6,7}
{4,5,6,7}
The a(0) = 0 through a(7) = 13 set-systems:
. . . . . {2}{12}{3}{13} {2}{3}{23} {2}{3}{23}
{2}{12}{3}{13} {2}{12}{3}{13}
{12}{3}{13}{23} {12}{3}{13}{23}
{2}{12}{13}{23} {2}{12}{13}{23}
{2}{12}{3}{123}
{2}{3}{13}{123}
{12}{3}{13}{123}
{12}{3}{23}{123}
{2}{12}{13}{123}
{2}{12}{23}{123}
{2}{13}{23}{123}
{3}{13}{23}{123}
{12}{13}{23}{123}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
fasmin[y_]:=Complement[y, Union@@Table[Union[s, #]& /@ Rest[Subsets[Complement[Union@@y, s]]], {s, y}]];
Table[Length[fasmin[Select[Subsets[Range[2, n]], Select[Tuples[bpe/@#], UnsameQ@@#&]=={}&]]], {n, 0, 10}]
CROSSREFS
The version with ones allowed is A370642, minimal case of A370637.
This is the minimal case of A370643.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A370585 counts maximal choosable sets.
Cf. A072639, A140637, A326031, A355529, A367905, A368109, A370589, A370591, A370636, A370639, A370640.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 11 2024
STATUS
approved
A370643
Number of subsets of {2..n} such that it is not possible to choose a different binary index of each element.
+10
5
0, 0, 0, 0, 0, 1, 7, 23, 46, 113, 287, 680, 1546, 3374, 7191, 15008
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
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..15.
EXAMPLE
The a(0) = 0 through a(7) = 23 subsets:
. . . . . {2,3,4,5} {2,4,6} {2,4,6}
{2,3,4,5} {2,3,4,5}
{2,3,4,6} {2,3,4,6}
{2,3,5,6} {2,3,4,7}
{2,4,5,6} {2,3,5,6}
{3,4,5,6} {2,3,5,7}
{2,3,4,5,6} {2,3,6,7}
{2,4,5,6}
{2,4,5,7}
{2,4,6,7}
{2,5,6,7}
{3,4,5,6}
{3,4,5,7}
{3,4,6,7}
{3,5,6,7}
{4,5,6,7}
{2,3,4,5,6}
{2,3,4,5,7}
{2,3,4,6,7}
{2,3,5,6,7}
{2,4,5,6,7}
{3,4,5,6,7}
{2,3,4,5,6,7}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Select[Subsets[Range[2, n]], Select[Tuples[bpe/@#], UnsameQ@@#&]=={}&]], {n, 0, 10}]
CROSSREFS
The case with ones allowed is A370637, differences A370589.
The minimal case is A370644, with ones A370642.
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. A072639, A326031, A355740, A367905, A368109.
Cf. A133686, A140637, A355529, A367867, A370583, A370636, A370640.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 10 2024
STATUS
approved
A370588
Number of subsets of {1..n} containing n such that only one set can be obtained by choosing a different prime factor of each element.
+10
3
0, 0, 1, 2, 2, 6, 6, 18, 12, 20, 36, 104, 76, 284, 320, 408
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3), so {4,5,6} is counted under a(6).
LINKS
Table of n, a(n) for n=0..15.
EXAMPLE
The a(0) = 0 through a(8) = 12 subsets:
. . {2} {3} {4} {5} {2,6} {7} {8}
{2,3} {3,4} {2,5} {3,6} {2,7} {3,8}
{3,5} {4,6} {3,7} {5,8}
{4,5} {2,5,6} {4,7} {6,8}
{2,3,5} {3,5,6} {5,7} {7,8}
{3,4,5} {4,5,6} {2,3,7} {3,5,8}
{2,5,7} {3,7,8}
{2,6,7} {5,6,8}
{3,4,7} {5,7,8}
{3,5,7} {6,7,8}
{3,6,7} {3,5,7,8}
{4,5,7} {5,6,7,8}
{4,6,7}
{2,3,5,7}
{2,5,6,7}
{3,4,5,7}
{3,5,6,7}
{4,5,6,7}
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], MemberQ[#, n] && Length[Select[Tuples[If[#==1, {}, First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==1&]], {n, 0, 10}]
CROSSREFS
First differences of A370584, cf. A370582, complement A370583.
For any number of choices we have A370586, complement A370587.
For binary indices see A370638, A370639, complement A370589.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370585 counts maximal choosable sets.
A370592 counts choosable partitions, complement A370593.
A370636 counts choosable subsets for binary indices, complement A370637.
Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367771, A367905.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 28 2024
STATUS
approved
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