cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A370644 Number of minimal subsets of {2..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 13, 13, 26, 56, 126, 243, 471, 812, 1438
Offset: 0

Views

Author

Gus Wiseman, Mar 11 2024

Keywords

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.

Examples

			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}

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.

Programs

  • 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}]

A370643 Number of subsets of {2..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 7, 23, 46, 113, 287, 680, 1546, 3374, 7191, 15008, 30016, 61013, 124354, 252577, 511229, 1031064, 2074281, 4164716, 8350912, 16729473, 33494928, 67034995, 134127390, 268325204, 536737665, 1073581062, 2147162124, 4294458549, 8589210382, 17178890873
Offset: 0

Views

Author

Gus Wiseman, Mar 10 2024

Keywords

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.

Examples

			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}

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.

Programs

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

Extensions

More terms from Jinyuan Wang, Mar 28 2025

A370807 Number of integer partitions of n into parts > 1 such that it is not possible to choose a different prime factor of each part.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 1, 4, 4, 8, 9, 15, 17, 25, 30, 43, 54, 72, 87, 115, 139, 181, 224, 283, 342, 429, 519, 647, 779, 967
Offset: 0

Views

Author

Gus Wiseman, Mar 04 2024

Keywords

Examples

			The a(0) = 0 through a(11) = 9 partitions:
  .  .  .  .  (22)  .  (33)   (322)  (44)    (333)   (55)     (443)
                       (42)          (332)   (432)   (82)     (533)
                       (222)         (422)   (522)   (433)    (542)
                                     (2222)  (3222)  (442)    (632)
                                                     (622)    (722)
                                                     (3322)   (3332)
                                                     (4222)   (4322)
                                                     (22222)  (5222)
                                                              (32222)

Crossrefs

These partitions are ranked by the odd terms of A355529, complement A368100.
The version for set-systems is A367903, complement A367902.
The version for factorizations is A368413, complement A368414.
With ones allowed we have A370593, complement A370592.
For a unique choice we have A370594, ranks A370647.
The version for divisors instead of factors is A370804, complement A370805.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts condensed partitions, ranks A368110.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A133686, A355739, A355740, A367771, A367867, A367905, A370583, A370585, A370586, A370636.

Programs

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

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

Views

Author

Gus Wiseman, Feb 28 2024

Keywords

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).

Examples

			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}

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.

Programs

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

Extensions

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

Views

Author

Gus Wiseman, Feb 28 2024

Keywords

Comments

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

Examples

			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

Crossrefs

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.

Programs

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

Extensions

More terms from Jinyuan Wang, Feb 14 2025

A370646 Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 62, 165, 475, 1400, 4334
Offset: 0

Views

Author

Gus Wiseman, Mar 12 2024

Keywords

Comments

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements.

Examples

			The multiset partition {{3},{1,3},{2,3}} has unique choice (3,1,2) so is counted under a(5).
Representatives of the a(1) = 1 through a(5) = 23 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}
       {1}{2}  {1}{22}    {1}{122}      {11}{122}
               {2}{12}    {11}{22}      {1}{1222}
               {1}{2}{3}  {12}{12}      {11}{222}
                          {1}{222}      {12}{122}
                          {12}{22}      {1}{2222}
                          {2}{122}      {12}{222}
                          {1}{2}{33}    {2}{1122}
                          {1}{3}{23}    {2}{1222}
                          {1}{2}{3}{4}  {22}{122}
                                        {1}{2}{233}
                                        {1}{22}{33}
                                        {1}{23}{23}
                                        {1}{2}{333}
                                        {1}{23}{33}
                                        {1}{3}{233}
                                        {2}{12}{33}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {1}{2}{3}{44}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

Crossrefs

For existence we have A368098, complement A368097.
Multisets of this type are ranked by A368101, see also A368100, A355529.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
Partitions of this type are counted by A370594, see also A370592, A370593.
Subsets of this type are also counted by A370638, see also A370636, A370637.
Factorizations of this type are A370645, see also A368414, A368413.
Set-systems of this type are A370818, see also A367902, A367903.
A000110 counts set partitions, non-isomorphic A000041.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A000612, A055621, A283877, A300913, A302545, A316983, A319616, A330223, A368095, A368412, A368422.

A370819 Number of subsets of {1..n-1} whose cardinality is one less than the length of the binary expansion of n; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 10, 15, 35, 56, 84, 120, 165, 220, 286, 364, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757, 658008, 749398, 850668
Offset: 0

Views

Author

Gus Wiseman, Mar 11 2024

Keywords

Examples

			The a(1) = 1 through a(7) = 15 subsets:
  {}  {1}  {1}  {1,2}  {1,2}  {1,2}  {1,2}
           {2}  {1,3}  {1,3}  {1,3}  {1,3}
                {2,3}  {1,4}  {1,4}  {1,4}
                       {2,3}  {1,5}  {1,5}
                       {2,4}  {2,3}  {1,6}
                       {3,4}  {2,4}  {2,3}
                              {2,5}  {2,4}
                              {3,4}  {2,5}
                              {3,5}  {2,6}
                              {4,5}  {3,4}
                                     {3,5}
                                     {3,6}
                                     {4,5}
                                     {4,6}
                                     {5,6}

Crossrefs

The version without subtracting one is A357812.
Dominates A370641, see also A370640.
A007318 counts subsets by cardinality.
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. A029837, A113473, A326031, A367905, A368109, A370636.

Programs

  • Mathematica
    Table[If[n==0,0,Binomial[n-1,IntegerLength[n,2]-1]],{n,0,15}]

Formula

a(n) = binomial(n - 1, A029837(n+1) - 1) = binomial(n - 1, A113473(n) - 1) = binomial(n - 1, A070939(n) - 1) for n > 0.
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