A326020 -id:A326020 - cp's OEIS frontend

A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.

0, 0, 1, 2, 4, 5, 9, 11, 17, 21, 29, 36, 50, 60, 78, 95, 123, 147, 185, 221, 274, 325, 399, 472, 574, 672, 810, 945, 1131, 1316, 1557, 1812, 2137, 2462, 2892, 3322, 3881, 4460, 5176, 5916, 6846, 7817, 8993, 10250, 11765, 13333, 15280, 17308, 19731, 22306

Offset: 0

Gus Wiseman, Aug 23 2023

nonn

Sets of this type may be called "positive combination-full".

Also subsets of {1..n} containing n whose greatest element can be written as a positive linear combination of the others.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(7) = 11 subsets:
  .  .  {1,2}  {1,3}    {1,4}    {1,5}    {1,6}      {1,7}
               {1,2,3}  {2,4}    {1,2,5}  {2,6}      {1,2,7}
                        {1,2,4}  {1,3,5}  {3,6}      {1,3,7}
                        {1,3,4}  {1,4,5}  {1,2,6}    {1,4,7}
                                 {2,3,5}  {1,3,6}    {1,5,7}
                                          {1,4,6}    {1,6,7}
                                          {1,5,6}    {2,3,7}
                                          {2,4,6}    {2,5,7}
                                          {1,2,3,6}  {3,4,7}
                                                     {1,2,3,7}
                                                     {1,2,4,7}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The nonnegative complement is A124506, first differences of A326083.
The binary complement is A288728, first differences of A007865.
First differences of A365043.
The complement is counted by A365045, first differences of A365044.
The nonnegative version is A365046, first differences of A364914.
Without re-usable parts we have A365069, first differences of A364534.
The binary version is A365070, first differences of A093971.
A085489 and A364755 count subsets with no sum of two distinct elements.
A088314 counts sets that can be linearly combined to obtain n.
A088809 and A364756 count subsets with some sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Or@@Table[combp[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]

a(n) = A088314(n) - 1.

A326036 Number of uniform complete integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 2, 2, 2, 2, 6, 3, 3, 5, 5, 3, 8, 5, 11, 10, 10, 9, 19, 13, 15, 17, 21, 18, 35, 26, 39, 40, 50, 50, 77, 63, 84, 88, 113, 103, 146, 132, 171, 180, 212, 214, 292, 276, 345, 363, 435, 442, 561, 569, 694, 729, 853, 891, 1108

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition of n is uniform if all parts appear with the same multiplicity, and complete if every nonnegative integer up to n is the sum of some submultiset.

			The initial terms count the following partitions:
   0: ()
   1: (1)
   2: (11)
   3: (21)
   3: (111)
   4: (1111)
   5: (11111)
   6: (321)
   6: (2211)
   6: (111111)
   7: (421)
   7: (1111111)
   8: (3311)
   8: (11111111)
   9: (222111)
   9: (111111111)
  10: (4321)
  10: (1111111111)
  11: (5321)
  11: (11111111111)

Cf. A002033, A047966, A072774, A108917, A126796, A188431, A276024.

Cf. A325781, A325791, A325988, A326020, A326035, A326037.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#]&&Sort[sums[Sort[#]]]==Range[Total[#]]&]],{n,0,30}]

A326114 Number of subsets of {2..n} containing no product of two or more (not necessarily distinct) elements.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 22, 44, 76, 116, 222, 444, 788, 1576, 3068, 5740, 8556, 17112, 31752, 63504, 116176, 221104, 438472, 876944, 1569424, 2447664, 4869576, 9070920, 17022360, 34044720, 61923312, 123846624, 234698720, 462007072, 922838192, 1734564112, 2591355792, 5182711584

Offset: 0

Gus Wiseman, Jun 06 2019

nonn

The strict case is A326117.
Also the number of subsets of {2..n} containing all of their integer products <= n. For example, the a(1) = 1 through a(5) = 12 subsets are:
  {}  {}  {}   {}     {}       {}
          {2}  {2}    {3}      {3}
               {3}    {4}      {4}
               {2,3}  {2,4}    {5}
                      {3,4}    {2,4}
                      {2,3,4}  {3,4}
                               {3,5}
                               {4,5}
                               {2,3,4}
                               {2,4,5}
                               {3,4,5}
                               {2,3,4,5}

			The a(1) = 1 through a(5) = 12 subsets:
  {}  {}   {}     {}     {}
      {2}  {2}    {2}    {2}
           {3}    {3}    {3}
           {2,3}  {4}    {4}
                  {2,3}  {5}
                  {3,4}  {2,3}
                         {2,5}
                         {3,4}
                         {3,5}
                         {4,5}
                         {2,3,5}
                         {3,4,5}

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326116, A326117.

a(n > 0) = A326076(n)/2.

Terms a(21) and beyond from Andrew Howroyd, Aug 30 2019

A326115 Number of maximal double-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 12, 12, 12, 12, 24, 24, 32, 32, 64, 64, 64, 64, 128, 128, 192, 192, 384, 384, 384, 384, 768, 768, 960, 960, 1920, 1920, 1920, 1920, 3840, 3840, 5760, 5760, 11520, 11520, 11520, 11520, 23040, 23040, 30720, 30720

Offset: 0

Gus Wiseman, Jun 06 2019

nonn

A set is double-free if no element is twice any other element.

			The a(1) = 1 through a(9) = 6 sets:
  {1}  {1}  {13}  {23}   {235}   {235}   {2357}   {13457}  {134579}
       {2}  {23}  {134}  {1345}  {256}   {2567}   {13578}  {135789}
                                 {1345}  {13457}  {14567}  {145679}
                                 {1456}  {14567}  {15678}  {156789}
                                                  {23578}  {235789}
                                                  {25678}  {256789}

Charlie Neder, Table of n, a(n) for n = 0..1000

Cf. A000931, A007865, A050291, A103580, A120641, A308546, A320340, A323092.

Cf. A325865, A326020, A326022, A326083.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,2*#]=={}&]]],{n,0,10}]

From Charlie Neder, Jun 11 2019: (Start)
a(n) = Product {k < n/2} A000931(8+floor(log_2(n/(2k+1)))).
a(2k+1) = a(2k), a(8k+4) = a(8k+3). (End)

a(16)-a(49) from Charlie Neder, Jun 11 2019

A308542 Number of subsets of {2..n} containing the product of any set of distinct elements whose product is <= n.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 56, 100, 200, 364, 728, 1184, 2368, 4448, 8056, 15008, 30016, 52736, 105472, 183424, 339840, 663616, 1327232, 2217088, 4434176, 8744320, 16559168, 30034624, 60069248, 103402112, 206804224, 379941440, 730800064, 1454649248, 2659869664, 4786282208

Offset: 1

Gus Wiseman, Jun 06 2019

nonn

First differs from A326116 at a(12) = 1184, A326116(12) = 1232.

If this sequence counts product-closed sets, A326116 counts product-free sets.

			The a(6) = 28 sets:
  {}  {2}  {2,4}  {2,3,6}  {2,3,4,6}  {2,3,4,5,6}
      {3}  {2,5}  {2,4,5}  {2,3,5,6}
      {4}  {2,6}  {2,4,6}  {2,4,5,6}
      {5}  {3,4}  {2,5,6}  {3,4,5,6}
      {6}  {3,5}  {3,4,5}
           {3,6}  {3,4,6}
           {4,5}  {3,5,6}
           {4,6}  {4,5,6}
           {5,6}

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326116, A326117.

Table[Length[Select[Subsets[Range[2,n]],SubsetQ[#,Select[Times@@@Subsets[#,{2}],#<=n&]]&]],{n,0,10}]

For n > 0, a(n) = A326081(n)/2.

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

A326021 Number of complete subsets of {1..n} with maximum n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 23, 45, 90, 180, 359, 717, 1432, 2862, 5723, 11444, 22887, 45772, 91541, 183078, 366151, 732295, 1464583, 2929158, 5858307, 11716603, 23433196, 46866379, 93732744, 187465471, 374930922, 749861819, 1499723610

Offset: 1

Gus Wiseman, Jun 04 2019

nonn

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

			The a(1) = 1 through a(7) = 12 subsets:
  {1}  {1,2}  {1,2,3}  {1,2,4}    {1,2,3,5}    {1,2,3,6}      {1,2,3,7}
                       {1,2,3,4}  {1,2,4,5}    {1,2,4,6}      {1,2,4,7}
                                  {1,2,3,4,5}  {1,2,3,4,6}    {1,2,3,4,7}
                                               {1,2,3,5,6}    {1,2,3,5,7}
                                               {1,2,4,5,6}    {1,2,3,6,7}
                                               {1,2,3,4,5,6}  {1,2,4,5,7}
                                                              {1,2,4,6,7}
                                                              {1,2,3,4,5,7}
                                                              {1,2,3,4,6,7}
                                                              {1,2,3,5,6,7}
                                                              {1,2,4,5,6,7}
                                                              {1,2,3,4,5,6,7}

Charlie Neder, Table of n, a(n) for n = 1..300
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326022, A326036.

Table[Length[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,10}]

a(18)-a(34) from Charlie Neder, Jun 05 2019

A326022 Number of minimal complete subsets of {1..n} with maximum n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 4, 8, 8, 8, 10, 14, 25, 40, 49, 62

Offset: 1

Gus Wiseman, Jun 04 2019

A set of positive integers summing to m is complete if every nonnegative integer up to m is the sum of some subset. For example, (1,2,3,6,13) is a complete set because we have:
= (empty sum)
= 1
= 2
= 3
= 1 + 3
= 2 + 3
= 6
= 6 + 1
= 6 + 2
= 6 + 3
= 1 + 3 + 6
= 2 + 3 + 6
= 1 + 2 + 3 + 6
and the remaining numbers 13-25 are obtained by adding 13 to each of these.

			The a(3) = 1 through a(9) = 8 subsets:
  {1,2,3}  {1,2,4}  {1,2,3,5}  {1,2,3,6}  {1,2,3,7}  {1,2,4,8}    {1,2,3,4,9}
                    {1,2,4,5}  {1,2,4,6}  {1,2,4,7}  {1,2,3,5,8}  {1,2,3,5,9}
                                                     {1,2,3,6,8}  {1,2,3,6,9}
                                                     {1,2,3,7,8}  {1,2,3,7,9}
                                                                  {1,2,4,5,9}
                                                                  {1,2,4,6,9}
                                                                  {1,2,4,7,9}
                                                                  {1,2,4,8,9}

Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326021, A326036.

fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]]],{n,10}]

A326037 Heinz numbers of uniform perfect integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 32, 42, 64, 100, 128, 256, 512, 798, 1024, 2048, 2744, 4096, 8192, 16384, 32768, 42294, 52900, 65536

Offset: 1

Gus Wiseman, Jun 04 2019

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition of n is uniform if all parts appear with the same multiplicity, and perfect if every nonnegative integer up to n is the sum of a unique submultiset.
The enumeration of these partitions by sum is given by A089723.

			The sequence of all uniform perfect integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (11)
      6: (21)
      8: (111)
     16: (1111)
     32: (11111)
     42: (421)
     64: (111111)
    100: (3311)
    128: (1111111)
    256: (11111111)
    512: (111111111)
    798: (8421)
   1024: (1111111111)
   2048: (11111111111)
   2744: (444111)
   4096: (111111111111)
   8192: (1111111111111)
  16384: (11111111111111)

Cf. A002033, A047966, A070941, A072774, A108917, A126796, A276024, A299702.

Cf. A325780, A325781, A326020, A326035, A326036.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],SameQ@@Last/@FactorInteger[#]&&Sort[hwt/@Divisors[#]]==Range[0,hwt[#]]&]

Intersection of A072774 (uniform), A299702 (knapsack), and A325781 (complete).

A358392 Number of nonempty subsets of {1, 2, ..., n} with GCD equal to 1 and containing the sum of any two elements whenever it is at most n.

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 19, 27, 46, 63, 113, 148, 253, 345, 539, 734, 1198, 1580, 2540, 3417, 5233, 7095, 11190, 14720, 22988, 31057, 47168, 63331, 98233, 129836, 200689, 269165, 406504, 546700, 838766, 1108583, 1700025, 2281517, 3437422, 4597833, 7023543, 9308824, 14198257, 18982014, 28556962

Offset: 1

Max Alekseyev, Nov 13 2022

nonn

Also, the number of distinct numerical semigroups that are generated by some subset of {1, 2, ..., n} and have a finite complement in the positive integers.

Max Alekseyev, Table of n, a(n) for n = 1..100
Marcel K. Goh et al., Counting numerical semigroups by largest element of minimal generating set, MathOverflow, 2022.

Inverse Moebius transform of A103580.

Cf. A007865, A050291, A051026, A085489, A139384, A151897, A308546, A326020, A326076, A326080, A326083, A326114.

a(n) = Sum_{k=1..n} moebius(k) * A103580(floor(n/k)).

A364841 Number of subsets S of {1..n} containing no element equal to the sum of a k-multiset of elements of S, for any 2 <= k <= |S|.

Original entry on oeis.org

1, 2, 3, 6, 9, 15, 21, 34, 49, 75, 105

Offset: 0

Gus Wiseman, Aug 15 2023

			The a(0) = 1 through a(5) = 15 subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {1,3}  {4}    {4}
                {2,3}  {1,3}  {5}
                       {1,4}  {1,3}
                       {2,3}  {1,4}
                       {3,4}  {1,5}
                              {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {3,4,5}

Cf. A007865, A085489, A103580, A151897, A236912, A326020, A326080, A326083, A364349, A364534.

Table[Length[Select[Subsets[Range[n]], Intersection[#,Join@@Table[Total/@Tuples[#,k], {k,2,Length[#]}]]=={}&]],{n,0,10}]

cp's OEIS Frontend

A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.

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A326036 Number of uniform complete integer partitions of n.

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A326114 Number of subsets of {2..n} containing no product of two or more (not necessarily distinct) elements.

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A326115 Number of maximal double-free subsets of {1..n}.

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A308542 Number of subsets of {2..n} containing the product of any set of distinct elements whose product is <= n.

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A326021 Number of complete subsets of {1..n} with maximum n.

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A326022 Number of minimal complete subsets of {1..n} with maximum n.

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A326037 Heinz numbers of uniform perfect integer partitions.

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