A370638 -id:A370638 - cp's OEIS frontend

A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

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

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) = 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}

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

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.

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) = A367902(n).

Partial sums of A370639.

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

A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 18, 36, 48, 68, 104, 208, 284, 568, 888, 1296, 1548, 3096, 3968, 7936, 10736, 15440, 24008, 48016, 58848, 73680, 114368, 132608, 176240, 352480, 449824, 899648, 994976, 1399968, 2160720, 2859584, 3296048, 6592096, 10156672, 14214576, 16892352

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3).

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {2,3}
                             {2,5}    {2,5}
                             {3,4}    {2,6}
                             {3,5}    {3,4}
                             {4,5}    {3,5}
                             {2,3,5}  {3,6}
                             {3,4,5}  {4,5}
                                      {4,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

For divisors instead of factors we have A051026, cf. A368110, A355740.
The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
For existence we have A370582, differences A370586.
For nonexistence we have A370583, differences A370587.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, cf. A370592, A370593.
For binary indices instead of factors we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
For unlabeled multiset partitions we have A370646, cf. A368098, A368097.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000720, A003963, A005117, A045778, A133686, A307984, A355739, A355744, A355745, A367905.

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

More terms from Jinyuan Wang, Mar 28 2025

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

A370594 Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 3, 4, 5, 5, 8, 10, 11, 7, 14, 13, 19, 23, 24, 20, 30, 33, 40, 47, 49, 55, 53, 72, 80, 90, 92, 110, 110, 132, 154, 169, 180, 201, 218, 246, 281, 302, 323, 348, 396, 433, 482, 530, 584, 618, 670, 754, 823, 903, 980, 1047, 1137

Offset: 0

Gus Wiseman, Feb 29 2024

nonn

			The partition (10,6,4) has unique choice (5,3,2) so is counted under a(20).
The a(0) = 1 through a(12) = 5 partitions:
()  .  (2)  (3)  (4)  (5)    .  (7)    (8)    (9)    (6,4)    (11)   (6,6)
                      (3,2)     (4,3)  (5,3)  (5,4)  (7,3)    (7,4)  (7,5)
                                (5,2)  (6,2)  (6,3)  (5,3,2)  (8,3)  (10,2)
                                              (7,2)           (9,2)  (5,4,3)
                                                                     (7,3,2)

The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
The version for subsets is A370584, cf. A370582, A370583, A370586, A370587.
Maximal sets of this type are counted by A370585.
For existence we have A370592.
For nonexistence we have A370593.
For divisors instead of factors we have A370595.
For subsets and binary indices we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
These partitions have ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000041, A000720, A003963, A355739, A355740, A367905, A368110.

Table[Length[Select[IntegerPartitions[n], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,30}]

A370640 Number of maximal subsets of {1..n} such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

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, 53329, 66993, 82811, 101963, 124381, 151286, 151286, 324695, 526866, 764438, 1038089, 1358129, 1725921, 2154668, 2640365, 3202985

Offset: 0

Gus Wiseman, Mar 10 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.

Also choices of A070939(n) elements of {1..n} such that it is possible to choose a different binary index of each.

			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}

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.

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}]

lista(nn) = my(b, m=Map(Mat([[[]], 1])), t, u, v, w, z); for(n=0, nn, t=Mat(m)~; b=Vecrev(binary(n)); u=select(i->b[i], [1..#b]); for(i=1, #t, v=t[1, i]; w=List([]); for(j=1, #v, for(k=1, #u, if(!setsearch(v[j], u[k]), listput(w, setunion(v[j], [u[k]]))))); w=Set(w); if(#w, z=0; mapisdefined(m, w, &z); mapput(m, w, z+t[2, i]))); print1(mapget(m, [[1..#b]]), ", ")); \\ Jinyuan Wang, Mar 28 2025

More terms from Jinyuan Wang, Mar 28 2025

A370639 Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

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

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(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}

Wikipedia, Axiom of choice.

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.

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

First differences of A370636.

a(19)-a(42) from Alois P. Heinz, Mar 09 2024

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

A370642 Number of minimal 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, 1, 3, 9, 26, 26, 40, 82, 175, 338, 636, 1114

Offset: 0

Gus Wiseman, Mar 10 2024

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(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}

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.

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}]

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}]

A370647 Numbers such that only one set can be obtained by choosing a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 23, 31, 33, 35, 39, 41, 51, 53, 55, 59, 65, 67, 69, 77, 83, 85, 87, 91, 93, 95, 97, 103, 109, 111, 119, 123, 127, 129, 131, 155, 157, 161, 165, 169, 177, 179, 183, 185, 187, 191, 201, 203, 205, 209, 211, 213, 217, 227, 235, 237, 241

Offset: 1

Gus Wiseman, Mar 06 2024

nonn

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.

			The prime indices of 91 are {4,6}, with only choice {2,3}, so 91 is in the sequence.
The terms together with their prime indices begin:
     1: {}        53: {16}      109: {29}
     3: {2}       55: {3,5}     111: {2,12}
     5: {3}       59: {17}      119: {4,7}
     7: {4}       65: {3,6}     123: {2,13}
    11: {5}       67: {19}      127: {31}
    15: {2,3}     69: {2,9}     129: {2,14}
    17: {7}       77: {4,5}     131: {32}
    19: {8}       83: {23}      155: {3,11}
    23: {9}       85: {3,7}     157: {37}
    31: {11}      87: {2,10}    161: {4,9}
    33: {2,5}     91: {4,6}     165: {2,3,5}
    35: {3,4}     93: {2,11}    169: {6,6}
    39: {2,6}     95: {3,8}     177: {2,17}
    41: {13}      97: {25}      179: {41}
    51: {2,7}    103: {27}      183: {2,18}

For nonexistence we have A355529, count A370593.
For binary instead of prime indices we have A367908, counted by A367904.
For existence we have A368100, count A370592.
For a sequence instead of set of factors we have A368101.
The version for subsets is A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
Partitions of this type are counted by A370594.
For subsets and binary indices we have A370638.
The version for factorizations is A370645, see also A368414, A368413.
For divisors instead of factors we have A370810, counted by A370595.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000720, A003963, A355739, A355740, A355744, A355745, A368110.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Union[Sort /@ Select[Tuples[prix/@prix[#]],UnsameQ@@#&]]]==1&]

cp's OEIS Frontend

A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

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A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

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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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A370594 Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.

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A370640 Number of maximal subsets of {1..n} such that it is possible to choose a different binary index of each element.

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A370639 Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

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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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A370642 Number of minimal subsets of {1..n} such that it is not possible to choose a different binary index of each element.

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