A368098 -id:A368098 - cp's OEIS frontend

A368422 Number of non-isomorphic set multipartitions of weight n satisfying a strict version of the axiom of choice.

1, 1, 2, 4, 9, 18, 43, 95, 233, 569

Offset: 0

Gus Wiseman, Dec 26 2023

A set multipartition is a finite multiset of finite nonempty sets. The weight of a set multipartition is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 set multipartitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}        {{1,2,3,4,5}}
         {{1},{2}}  {{1},{2,3}}    {{1,2},{1,2}}      {{1},{2,3,4,5}}
                    {{2},{1,2}}    {{1},{2,3,4}}      {{1,2},{3,4,5}}
                    {{1},{2},{3}}  {{1,2},{3,4}}      {{1,4},{2,3,4}}
                                   {{1,3},{2,3}}      {{2,3},{1,2,3}}
                                   {{3},{1,2,3}}      {{4},{1,2,3,4}}
                                   {{1},{2},{3,4}}    {{1},{2,3},{2,3}}
                                   {{1},{3},{2,3}}    {{1},{2},{3,4,5}}
                                   {{1},{2},{3},{4}}  {{1},{2,3},{4,5}}
                                                      {{1},{2,4},{3,4}}
                                                      {{1},{4},{2,3,4}}
                                                      {{2},{1,3},{2,3}}
                                                      {{2},{3},{1,2,3}}
                                                      {{3},{1,3},{2,3}}
                                                      {{4},{1,2},{3,4}}
                                                      {{1},{2},{3},{4,5}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{3},{4},{5}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A134964, complement A140637.
Set multipartitions have ranks A302478, cf. A073576.
The case of labeled graphs is A133686, complement A367867.
The complement without repeats is A368094 connected A368409.
Without repeats we have A368095, connected A368410.
The complement allowing repeats is A368097, ranks A355529.
Allowing repeated elements gives A368098, ranks A368100.
Factorizations of this type are counted by A368414, complement A368413.
The complement is counted by A368421.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A306005, A317533, A318360, A367903.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A370589 Number of subsets of {1..n} containing 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, 1, 6, 17, 42, 67, 175, 400, 870, 1841, 3820, 7837, 15920, 30997, 63370, 128348, 258699, 520042, 1043284, 2090732, 4186382, 8379022, 16765549, 33540664, 67092258, 134198633, 268412631, 536844414, 1073710403, 2147296425, 4294753612, 8589686922, 17179580003

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 binary indices of {1,4,5} are {{1},{3},{1,3}}, from which it is not possible to choose three different elements, so S is counted under a(3).
The binary indices of S = {1,6,8,9} are {{1},{2,3},{4},{1,4}}, from which it is not possible to choose four different elements, so S is counted under a(9).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,2,3,4}  {1,4,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {3,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

Wikipedia, Axiom of choice.

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 A370586, differences of A370582.
For prime indices we have A370587, differences of A370583.
Partial sums are A370637/A370643, minima A370642/A370644.
The complement is counted by A370639, partial sums A370636.
The version for a unique choice is A370641, partial sums 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, A072639, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.

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

a(19)-a(35) from Alois P. Heinz, Mar 09 2024

A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 15, 32, 80, 198, 528

Offset: 0

Gus Wiseman, Dec 25 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 set-systems:
  {1}  {12}  {123}    {1234}    {12345}      {123456}
             {2}{12}  {13}{23}  {14}{234}    {125}{345}
                      {3}{123}  {23}{123}    {134}{234}
                                {4}{1234}    {15}{2345}
                                {2}{13}{23}  {34}{1234}
                                {2}{3}{123}  {5}{12345}
                                {3}{13}{23}  {1}{14}{234}
                                             {12}{13}{23}
                                             {1}{23}{123}
                                             {13}{24}{34}
                                             {14}{24}{34}
                                             {3}{14}{234}
                                             {3}{23}{123}
                                             {3}{4}{1234}
                                             {4}{14}{234}

Wikipedia, Axiom of choice.

For unlabeled graphs we have A005703, connected case of A134964.
For labeled graphs we have A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
The complement without connectedness is A367903, ranks A367907.
Without connectedness we have A368095, ranks A367906,
Complement with repeats: A368097, connected case of A368411, ranks A355529.
The complement is counted by A368409, connected case of A368094.
With repeats allowed: A368412, connected case of A368098, ranks A368100.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A001055, A140637, A302545, A306005, A321194, A321405, A367902, A368414, A368422.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]}, {i,Length[p]}])],{p,Permutations[Union@@m]}]]];
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]]]]]]]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A368421 Number of non-isomorphic set multipartitions of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 2, 7, 16, 47, 116, 325, 861

Offset: 0

Gus Wiseman, Dec 26 2023

A set multipartition is a finite multiset of finite nonempty sets. The weight of a set multipartition is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any sequence of nonempty sets Y, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 16 set multipartitions:
  {{1},{1}}  {{1},{1},{1}}  {{1},{1},{2,3}}    {{1},{1},{2,3,4}}
             {{1},{2},{2}}  {{1},{2},{1,2}}    {{2},{1,2},{1,2}}
                            {{2},{2},{1,2}}    {{3},{3},{1,2,3}}
                            {{1},{1},{1},{1}}  {{1},{1},{1},{2,3}}
                            {{1},{1},{2},{2}}  {{1},{1},{3},{2,3}}
                            {{1},{2},{2},{2}}  {{1},{2},{2},{1,2}}
                            {{1},{2},{3},{3}}  {{1},{2},{2},{3,4}}
                                               {{1},{2},{3},{2,3}}
                                               {{1},{3},{3},{2,3}}
                                               {{2},{2},{2},{1,2}}
                                               {{1},{1},{1},{1},{1}}
                                               {{1},{1},{2},{2},{2}}
                                               {{1},{2},{2},{2},{2}}
                                               {{1},{2},{2},{3},{3}}
                                               {{1},{2},{3},{3},{3}}
                                               {{1},{2},{3},{4},{4}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A140637, complement A134964.
Set multipartitions have ranks A302478, cf. A073576.
The case of labeled graphs is A367867, complement A133686.
With distinct edges we have A368094 connected A368409.
The complement with distinct edges is A368095, connected A368410.
Allowing repeated elements gives A368097, ranks A355529.
The complement allowing repeats is A368098, ranks A368100.
Factorizations of this type are counted by A368413, complement A368414.
The complement is counted by A368422.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A306005, A316983, A317533, A318360, A367903, A367905, A367907.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 01 2024

nonn

All of these factorizations are co-balanced (A340596).

			The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
  (2)  (2*6)  (3*8)   (4*9)   (8*9)   (3*5*8)   (2*72)   (4*5*9)   (3*96)
       (3*4)  (4*6)   (6*6)   (2*36)  (4*5*6)   (3*48)   (5*6*6)   (4*72)
              (2*12)  (2*18)  (3*24)  (2*3*20)  (4*36)   (2*3*30)  (6*48)
                      (3*12)  (4*18)  (2*5*12)  (6*24)   (2*5*18)  (8*36)
                              (6*12)  (2*6*10)  (8*18)   (2*6*15)  (9*32)
                                      (3*4*10)  (9*16)   (2*9*10)  (12*24)
                                                (12*12)  (3*4*15)  (16*18)
                                                         (3*5*12)  (2*144)
                                                         (3*6*10)

Multisets of this type are ranked by A368101, see also A368100, A355529.
For nonexistence we have A368413, complement A368414.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, see also A370592, A370593.
Subsets of this type are counted by A370638, see also A370636, A370637.
For unlabeled multiset partitions we have A370646, also A368098, A368097.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A027746 lists prime factors, A112798 indices, length A001222.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
A355741 counts ways to choose a prime factor of each prime index.
For set-systems see A367902-A367908.
Cf. A000040, A000720, A003963, A340596, A340653, A355744, A355745, A368110.

facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Length[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]],{n,100}]

A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 2, 4, 11, 25, 75, 206, 650, 2049, 6895

Offset: 0

Gus Wiseman, Dec 26 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{2},{1,2}}  {{1,2,3,3}}
                               {{1,2,3,4}}
                               {{1},{1,2,2}}
                               {{1,2},{1,2}}
                               {{1,2},{2,2}}
                               {{1,3},{2,3}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
This is the connected case of A368098, ranks A368100.
Complement set-systems: A368409, connected case of A368094, ranks A367907.
For set-systems we have A368410, connected case of A368095, ranks A367906.
The complement is A368411, connected case of A368097, ranks A355529.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A140637, A302545, A316983, A317533, A319616, A367902, A367905, A368187.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
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]]]]]]]]];
Table[Length[Union[brute /@ Select[mpm[n],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A368411 Number of non-isomorphic connected multiset partitions of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 2, 6, 15, 50, 148, 509, 1725, 6218

Offset: 0

Gus Wiseman, Dec 26 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 15 multiset partitions:
  {{1},{1}}  {{1},{1,1}}    {{1},{1,1,1}}      {{1},{1,1,1,1}}
             {{1},{1},{1}}  {{1,1},{1,1}}      {{1,1},{1,1,1}}
                            {{1},{1},{1,1}}    {{1},{1},{1,1,1}}
                            {{1},{2},{1,2}}    {{1},{1,1},{1,1}}
                            {{2},{2},{1,2}}    {{1},{1},{1,2,2}}
                            {{1},{1},{1},{1}}  {{1},{1,2},{2,2}}
                                               {{1},{2},{1,2,2}}
                                               {{2},{1,2},{1,2}}
                                               {{2},{1,2},{2,2}}
                                               {{2},{2},{1,2,2}}
                                               {{3},{3},{1,2,3}}
                                               {{1},{1},{1},{1,1}}
                                               {{1},{2},{2},{1,2}}
                                               {{2},{2},{2},{1,2}}
                                               {{1},{1},{1},{1},{1}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A140638, connected case of A367867.
The complement for labeled graphs is A129271, connected case of A133686.
This is the connected case of A368097.
For set-systems we have A368409, connected case of A368094, ranks A367907.
Complement set-systems: A368410, connected case of A368095, ranks A367906.
The complement is A368412, connected case of A368098, ranks A368100.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A140637, A302545, A316983, A317533, A319616, A367903, A367905, A368187.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
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]]]]]]]]];
Table[Length[Union[brute /@ Select[mpm[n],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

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).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 4, 4, 7, 11, 16, 16, 30, 30, 39, 73

Offset: 0

Gus Wiseman, Feb 28 2024

			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}

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.

Table[Length[fasmin[Select[Subsets[Range[n]], Length[Select[Tuples[prix/@#],UnsameQ@@#&]]==0&]]], {n,0,15}]

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.

Original entry on oeis.org

0, 0, 1, 2, 2, 6, 6, 18, 12, 20, 36, 104, 76, 284, 320, 408, 252, 1548, 872, 3968, 2800, 4704, 8568, 24008, 10832, 14832, 40688, 18240, 43632, 176240, 97344, 449824, 95328, 404992, 760752, 698864, 436464, 3296048, 3564576, 4057904, 2677776, 16892352, 8676576

Offset: 0

Gus Wiseman, Feb 28 2024

nonn

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).

			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}

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.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==1&]],{n,0,10}]

More terms from Jinyuan Wang, Mar 28 2025

A370590 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 5, 2, 4, 14, 25, 13, 38, 46, 66, 28, 178, 57, 235, 106, 238, 656, 1235, 288, 445, 2192, 664, 2016, 6840, 2300, 9140, 888, 6236, 17692, 14724, 7320, 56000, 60472, 70252, 37160, 223884, 66428, 290312, 113172, 80544, 517392, 1001420, 114336

Offset: 0

Gus Wiseman, Feb 28 2024

nonn

For example, the set {4,7,9,10} has choice (2,7,3,5) so is counted under a(10).

			The a(0) = 0 through a(10) = 14 subsets (A = 10):
  .  .  2  23  34  235  256  2357  3578  2579  237A
                   345  356  2567  5678  4579  267A
                        456  3457        5679  279A
                             3567        5789  347A
                             4567              357A
                                               367A
                                               378A
                                               467A
                                               479A
                                               567A
                                               579A
                                               678A
                                               679A
                                               789A

Not requiring n gives A370585, maximal case of A370582, complement A370583.
Maximal case of A370586, complement A370587, unique A370588.
An opposite version is A370591.
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.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A133686, A355739, A355744, A355745, A367771, A370584, A370636, A370639.

Table[Length[Select[Subsets[Range[n],{PrimePi[n]}],MemberQ[#,n]&&Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#],UnsameQ@@#&]]>0&]],{n,0,10}]

More terms from Jinyuan Wang, Feb 14 2025

cp's OEIS Frontend

A368422 Number of non-isomorphic set multipartitions of weight n satisfying a strict version of the axiom of choice.

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A370589 Number of subsets of {1..n} containing n such that it is not possible to choose a different binary index of each element.

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A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

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A368421 Number of non-isomorphic set multipartitions of weight n contradicting a strict version of the axiom of choice.

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A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

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A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

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A368411 Number of non-isomorphic connected multiset partitions of weight n contradicting a strict version of the axiom of choice.

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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).

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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.