A326020 -id:A326020 - cp's OEIS frontend

A365380 Number of subsets of {1..n} that cannot be linearly combined using nonnegative coefficients to obtain n.

1, 1, 2, 2, 6, 4, 16, 12, 32, 32, 104, 48, 256, 208, 448, 448, 1568, 896, 3840, 2368, 6912, 7680, 22912, 10752, 50688, 44800, 104448, 88064, 324096, 165888, 780288, 541696, 1458176, 1519616, 4044800, 2220032, 10838016, 8744960, 20250624, 16433152, 62267392, 34865152

Offset: 1

Gus Wiseman, Sep 04 2023

nonn

			The set {4,5,6} cannot be linearly combined to obtain 7 so is counted under a(7), but we have 8 = 2*4 + 0*5 + 0*6, so it is not counted under a(8).
The a(1) = 1 through a(8) = 12 subsets:
  {}  {}  {}   {}   {}     {}     {}       {}
          {2}  {3}  {2}    {4}    {2}      {3}
                    {3}    {5}    {3}      {5}
                    {4}    {4,5}  {4}      {6}
                    {2,4}         {5}      {7}
                    {3,4}         {6}      {3,6}
                                  {2,4}    {3,7}
                                  {2,6}    {5,6}
                                  {3,5}    {5,7}
                                  {3,6}    {6,7}
                                  {4,5}    {3,6,7}
                                  {4,6}    {5,6,7}
                                  {5,6}
                                  {2,4,6}
                                  {3,5,6}
                                  {4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 1..100
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The complement is counted by A365073, without n A365542.
The binary complement is A365314, positive A365315.
The binary case is A365320, positive A365321.
For positive coefficients we have A365322, complement A088314.
A124506 appears to count combination-free subsets, differences of A326083.
A179822 counts sum-closed subsets, first differences of A326080.
A288728 counts binary sum-free subsets, first differences of A007865.
A365046 counts combination-full subsets, first differences of A364914.
A365071 counts sum-free subsets, first differences of A151897.
Cf. A050291, A085489, A088528, A088809, A093971, A326020, A364350, A364534, A365043, A365045.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n-1]],combs[n,#]=={}&]],{n,5}]

a(n) = 2^n - A365073(n).

Terms a(12) and beyond from Andrew Howroyd, Sep 04 2023

A367222 Number of subsets of {1..n} whose cardinality can be written as a nonnegative linear combination of the elements.

Original entry on oeis.org

1, 2, 3, 6, 12, 24, 49, 101, 207, 422, 859, 1747, 3548, 7194, 14565, 29452, 59496, 120086, 242185, 488035, 982672, 1977166, 3975508, 7989147, 16047464, 32221270, 64674453, 129775774, 260337978, 522124197, 1046911594, 2098709858, 4206361369, 8429033614, 16887728757, 33829251009, 67755866536, 135687781793, 271693909435

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			The set {1,2,4} has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(4).
The a(0) = 1 through a(4) = 12 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {1,2}  {1,2}    {1,2}
                  {1,3}    {1,3}
                  {2,3}    {1,4}
                  {1,2,3}  {2,3}
                           {2,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222*   A367223
  ranks:       A367224    A367225    A367226    A367227
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A326020 counts complete subsets.
A365046 counts combination-full subsets, differences of A364914.
Triangles:
A008284 counts partitions by length, strict A008289.
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts subsets containing two distinct elements summing to k.
Cf. A068911, A088314, A103580, A116861, A326080, A364350, A365073, A365311, A365376, A365380, A365544.

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

from itertools import combinations
from sympy.utilities.iterables import partitions
def A367222(n):
    c, mlist = 1, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(1,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for s in mlist[k-1]:
                if s <= ws:
                    c += 1
                    break
    return c # Chai Wah Wu, Nov 16 2023

a(n) = 2^n - A367223(n).

a(13)-a(33) from Chai Wah Wu, Nov 15 2023

a(34)-a(38) from Max Alekseyev, Feb 25 2025

A367223 Number of subsets of {1..n} whose cardinality cannot be written as a nonnegative linear combination of the elements.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 15, 27, 49, 90, 165, 301, 548, 998, 1819, 3316, 6040, 10986, 19959, 36253, 65904, 119986, 218796, 399461, 729752, 1333162, 2434411, 4441954, 8097478, 14746715, 26830230, 48773790, 88605927, 160900978, 292140427, 530487359, 963610200, 1751171679, 3183997509

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			3 cannot be written as a nonnegative linear combination of 2, 4, and 5, so {2,4,5} is counted under a(6).
The a(2) = 1 through a(6) = 15 subsets:
  {2}  {2}  {2}    {2}      {2}
       {3}  {3}    {3}      {3}
            {4}    {4}      {4}
            {3,4}  {5}      {5}
                   {3,4}    {6}
                   {3,5}    {3,4}
                   {4,5}    {3,5}
                   {2,4,5}  {3,6}
                            {4,5}
                            {4,6}
                            {5,6}
                            {2,4,5}
                            {2,4,6}
                            {2,5,6}
                            {4,5,6}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223*
  ranks:       A367224    A367225    A367226    A367227
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A365046 counts combination-full subsets, differences of A364914.
Triangles:
A116861 counts positive linear combinations of strict partitions of k.
A364916 counts linear combinations of strict partitions of k.
A366320 counts subsets without a subset summing to k, with A365381.
Cf. A068911, A088314, A103580, A237667, A326020, A326080, A364350, A365073, A365312, A365377, A365380.

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

from itertools import combinations
from sympy.utilities.iterables import partitions
def A367223(n):
    c, mlist = 0, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(1,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for s in mlist[k-1]:
                if s <= ws:
                    break
            else:
                c += 1
    return c # Chai Wah Wu, Nov 16 2023

a(n) = 2^n - A367222(n).

a(14)-a(33) from Chai Wah Wu, Nov 15 2023

a(34)-a(38) from Max Alekseyev, Feb 25 2025

A326076 Number of subsets of {1..n} containing all of their integer products <= n.

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 44, 88, 152, 232, 444, 888, 1576, 3152, 6136, 11480, 17112, 34224, 63504, 127008, 232352, 442208, 876944, 1753888, 3138848, 4895328, 9739152, 18141840, 34044720, 68089440, 123846624, 247693248, 469397440, 924014144, 1845676384, 3469128224, 5182711584

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

The strict case is A326081.

			The a(0) = 1 through a(4) = 12 sets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {3}
           {1,2}  {3}      {4}
                  {1,2}    {1,3}
                  {1,3}    {1,4}
                  {2,3}    {2,4}
                  {1,2,3}  {3,4}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}
The a(6) = 44 sets:
  {}  {1}  {1,3}  {1,2,4}  {1,2,4,5}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {3}  {1,4}  {1,3,4}  {1,2,4,6}  {1,2,4,5,6}
      {4}  {1,5}  {1,3,5}  {1,3,4,5}  {1,3,4,5,6}
      {5}  {1,6}  {1,3,6}  {1,3,4,6}  {2,3,4,5,6}
      {6}  {2,4}  {1,4,5}  {1,3,5,6}
           {3,4}  {1,4,6}  {1,4,5,6}
           {3,5}  {1,5,6}  {2,3,4,6}
           {3,6}  {2,4,5}  {2,4,5,6}
           {4,5}  {2,4,6}  {3,4,5,6}
           {4,6}  {3,4,5}
           {5,6}  {3,4,6}
                  {3,5,6}
                  {4,5,6}

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

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

a(n)={
    my(lim=vector(n, k, sqrtint(k)));
    my(accept(b, k)=for(i=2, lim[k], if(k%i ==0 && bittest(b, i) && bittest(b, k/i), return(0))); 1);
    my(recurse(k, b)=
      my(m=1);
      for(j=max(2*k, n\2+1), min(2*k+1, n), if(accept(b, j), m*=2));
      k++;
      m*if(k > n\2, 1, self()(k, b + (1<Andrew Howroyd, Aug 30 2019

a(n) = 2*A326114(n) for n > 0. - Andrew Howroyd, Aug 30 2019

a(16)-a(30) from Andrew Howroyd, Aug 16 2019

Terms a(31) and beyond from Andrew Howroyd, Aug 30 2019

A326117 Number of subsets of {1..n} containing no products of two or more distinct elements.

Original entry on oeis.org

1, 2, 3, 5, 9, 17, 29, 57, 101, 201, 365, 729, 1233, 2465, 4593, 8297, 15921, 31841, 55953, 111905, 195713, 362337, 697361, 1394721, 2334113, 4668225, 9095393, 17225313, 31242785, 62485569, 106668609, 213337217, 392606529, 755131841, 1491146913, 2727555425, 4947175713

Offset: 0

Gus Wiseman, Jun 06 2019

nonn

If this sequence counts product-free sets, A326081 counts product-closed sets.

			The a(6) = 28 sets:
  {}  {1}  {2,3}  {2,3,4}  {2,3,4,5}
      {2}  {2,4}  {2,3,5}  {2,4,5,6}
      {3}  {2,5}  {2,4,5}  {3,4,5,6}
      {4}  {2,6}  {2,4,6}
      {5}  {3,4}  {2,5,6}
      {6}  {3,5}  {3,4,5}
           {3,6}  {3,4,6}
           {4,5}  {3,5,6}
           {4,6}  {4,5,6}
           {5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..47

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326116, A308542.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Select[Times@@@Subsets[#,{2}],#<=n&]]=={}&]],{n,0,20}]

For n > 0, a(n) = A326116(n) + 1.

Terms a(21)-a(36) from Andrew Howroyd, Aug 30 2019

A308546 Number of double-closed subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 6, 8, 16, 24, 48, 60, 120, 180, 360, 480, 960, 1440, 2880, 3456, 6912, 10368, 20736, 27648, 55296, 82944, 165888, 207360, 414720, 622080, 1244160, 1658880, 3317760, 4976640, 9953280, 11612160, 23224320, 34836480, 69672960, 92897280

Offset: 0

Gus Wiseman, Jun 06 2019

nonn

These are subsets containing twice any element whose double is <= n.
Also the number of subsets of {1..n} containing half of every element that is even. For example, the a(6) = 24 subsets are:
  {}  {1}  {1,2}  {1,2,3}  {1,2,3,4}  {1,2,3,4,5}  {1,2,3,4,5,6}
      {3}  {1,3}  {1,2,4}  {1,2,3,5}  {1,2,3,4,6}
      {5}  {1,5}  {1,2,5}  {1,2,3,6}  {1,2,3,5,6}
           {3,5}  {1,3,5}  {1,2,4,5}
           {3,6}  {1,3,6}  {1,3,5,6}
                  {3,5,6}

			The a(6) = 24 subsets:
  {}  {4}  {2,4}  {1,2,4}  {1,2,4,5}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {5}  {3,6}  {2,4,5}  {1,2,4,6}  {1,2,4,5,6}
      {6}  {4,5}  {2,4,6}  {2,3,4,6}  {2,3,4,5,6}
           {4,6}  {3,4,6}  {2,4,5,6}
           {5,6}  {3,5,6}  {3,4,5,6}
                  {4,5,6}

Charlie Neder, Table of n, a(n) for n = 0..500

Cf. A007865, A050291, A103580, A120641, A320340, A323092, A325864, A326020, A326076, A326083, A326115.

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

From Charlie Neder, Jun 10 2019: (Start)
a(n) = Product_{k < n/2} (2 + floor(log_2(n/(2k+1)))).
a(0) = 1, a(n) = a(n-1) * (1 + 1/A001511(n)). (End)

a(21)-a(36) from Charlie Neder, Jun 10 2019

A365043 Number of subsets of {1..n} whose greatest element can be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 21, 32, 49, 70, 99, 135, 185, 245, 323, 418, 541, 688, 873, 1094, 1368, 1693, 2092, 2564, 3138, 3810, 4620, 5565, 6696, 8012, 9569, 11381, 13518, 15980, 18872, 22194, 26075, 30535, 35711, 41627, 48473, 56290, 65283, 75533, 87298, 100631, 115911, 133219

Offset: 0

Gus Wiseman, Aug 25 2023

nonn

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

Also subsets of {1..n} such that some element can be written as a (strictly) positive linear combination of the others.

			The subset S = {3,4,9} has 9 = 3*3 + 0*4, but this is not strictly positive, so S is not counted under a(9).
The subset S = {3,4,10} has 10 = 2*3 + 1*4, so S is counted under a(10).
The a(0) = 0 through a(5) = 12 subsets:
  .  .  {1,2}  {1,2}    {1,2}    {1,2}
               {1,3}    {1,3}    {1,3}
               {1,2,3}  {1,4}    {1,4}
                        {2,4}    {1,5}
                        {1,2,3}  {2,4}
                        {1,2,4}  {1,2,3}
                        {1,3,4}  {1,2,4}
                                 {1,2,5}
                                 {1,3,4}
                                 {1,3,5}
                                 {1,4,5}
                                 {2,3,5}

Bert Dobbelaere, Table of n, a(n) for n = 0..100
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The binary complement is A007865, first differences A288728.
The binary version is A093971, first differences A365070.
The nonnegative complement is A326083, first differences A124506.
The nonnegative version is A364914, first differences A365046.
First differences are A365042.
The complement is counted by A365044, first differences A365045.
Without re-usable parts we have A364534, first differences A365069.
A085489 and A364755 count subsets with no sum of two distinct elements.
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[Rest[Subsets[Range[n]]],combp[Last[#],Union[Most[#]]]!={}&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A365043(n):
    mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
    return sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023

a(n) = 2^n - A365044(n).

a(15)-a(35) from Chai Wah Wu, Nov 20 2023

More terms from Bert Dobbelaere, Apr 28 2025

A364755 Number of subsets of {1..n} containing n but not containing the sum of any two distinct elements.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 15, 24, 41, 60, 99, 149, 236, 355, 552, 817, 1275, 1870, 2788, 4167, 6243, 9098, 13433, 19718, 28771, 42137, 60652, 88603, 127555, 185200, 261781, 382931, 541022, 783862, 1096608, 1595829, 2217467, 3223064, 4441073, 6465800, 8893694

Offset: 0

Gus Wiseman, Aug 11 2023

nonn

			The subset S = {1,3,6,8} has pair-sums {4,7,9,11,14}, which are disjoint from S, so it is counted under a(8).
The a(1) = 1 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}
                     {2,3,4}  {1,2,5}  {5,6}
                              {1,3,5}  {1,2,6}
                              {2,4,5}  {1,3,6}
                              {3,4,5}  {1,4,6}
                                       {2,3,6}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}
                                       {4,5,6}
                                       {3,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..75

Partial sums are A085489(n) - 1, complement counted by A364534.
With re-usable parts we have A288728.
The complement with n is counted by A364756, first differences of A088809.
Cf. A007865, A050291, A054519, A093971, A151897, A236912, A326020, A326080, A326083, A364272, A364349, A364533.

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

First differences of A085489.

a(21) onwards added (using A085489) by Andrew Howroyd, Jan 13 2024

A365320 Number of pairs of distinct positive integers <= n that cannot be linearly combined with nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 7, 5, 12, 12, 27, 14, 42, 36, 47, 47, 83, 58, 109, 80, 116, 126, 172, 111, 195, 192, 219, 202, 294, 210, 342, 286, 354, 369, 409, 324, 509, 480, 523, 452, 640, 507, 711, 622, 675, 747, 865, 654, 916, 842, 964, 922, 1124, 940, 1147, 1029

Offset: 0

Gus Wiseman, Sep 06 2023

nonn

Are there only two cases of nonzero adjacent equal parts, at positions n = 9, 15?

			The pair p = (3,6) cannot be linearly combined to obtain 8 or 10, so p is counted under a(8) and a(10), but we have 9 = 1*3 + 1*6 or 9 = 3*3 + 0*6, so p not counted under a(9).
The a(5) = 2 through a(10) = 12 pairs:
  (2,4)  (4,5)  (2,4)  (3,6)  (2,4)  (3,6)
  (3,4)         (2,6)  (3,7)  (2,6)  (3,8)
                (3,5)  (5,6)  (2,8)  (3,9)
                (3,6)  (5,7)  (4,6)  (4,7)
                (4,5)  (6,7)  (4,7)  (4,8)
                (4,6)         (4,8)  (4,9)
                (5,6)         (5,6)  (6,7)
                              (5,7)  (6,8)
                              (5,8)  (6,9)
                              (6,7)  (7,8)
                              (6,8)  (7,9)
                              (7,8)  (8,9)

The unrestricted version is A000217, ranks A001358.
For strict partitions we have A365312, complement A365311.
The (binary) complement is A365314, positive A365315.
The case of positive coefficients is A365321, for all subsets A365322.
For partitions we have A365378, complement A365379.
For all subsets instead of just pairs we have A365380, complement A365073.
A004526 counts partitions of length 2, shift right for strict.
A007865 counts sum-free subsets, complement A093971.
A179822 and A326080 count sum-closed subsets.
A326083 and A124506 appear to count combination-free subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A070880, A088314, A088809, A151897, A326020, A364839.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}],combs[n,#]=={}&]],{n,0,30}]

from itertools import count
from sympy import divisors
def A365320(n):
    a = set()
    for i in range(1,n+1):
        if not n%i:
            a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
        else:
            for j in count(0,i):
                if j > n:
                    break
                k = n-j
                for d in divisors(k):
                    if d>=i:
                        break
                    a.add((d,i))
    return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 13 2023

A365322 Number of subsets of {1..n} that cannot be linearly combined using positive coefficients to obtain n.

Original entry on oeis.org

0, 1, 2, 5, 11, 26, 54, 116, 238, 490, 994, 2011, 4045, 8131, 16305, 32672, 65412, 130924, 261958, 524066, 1048301, 2096826, 4193904, 8388135, 16776641, 33553759, 67108053, 134216782, 268434324, 536869595, 1073740266, 2147481835, 4294965158, 8589932129

Offset: 0

Gus Wiseman, Sep 04 2023

nonn

We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.

			The set {1,3} has 4 = 1 + 3 so is not counted under a(4). However, 3 cannot be written as a linear combination of {1,3} using all positive coefficients, so it is counted under a(3).
The a(1) = 1 through a(4) = 11 subsets:
  {}  {}     {}       {}
      {1,2}  {2}      {3}
             {1,3}    {1,4}
             {2,3}    {2,3}
             {1,2,3}  {2,4}
                      {3,4}
                      {1,2,3}
                      {1,2,4}
                      {1,3,4}
                      {2,3,4}
                      {1,2,3,4}

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

The complement is counted by A088314.
The version for strict partitions is A088528.
The nonnegative complement is counted by A365073, without n A365542.
The binary complement is A365315, nonnegative A365314.
The binary version is A365321, nonnegative A365320.
For nonnegative coefficients we have A365380.
A085489 and A364755 count subsets without the sum of two distinct elements.
A124506 appears to count combination-free subsets, differences of A326083.
A179822 counts sum-closed subsets, first differences of A326080.
A364350 counts combination-free strict partitions, non-strict A364915.
A365046 counts combination-full subsets, first differences of A364914.
Cf. A070880, A088809, A093971, A151897, A326020, A364272, A364839, A365043, A365381.

b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x->{x[], i}, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
a:= n-> 2^n-nops(b(n$2)):
seq(a(n), n=0..33);  # Alois P. Heinz, Sep 04 2023

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

from sympy.utilities.iterables import partitions
def A365322(n): return (1<Chai Wah Wu, Sep 14 2023

a(n) = 2^n - A088314(n).

a(n) = A070880(n) + 2^(n-1) for n>=1.

More terms from Alois P. Heinz, Sep 04 2023

cp's OEIS Frontend

A365380 Number of subsets of {1..n} that cannot be linearly combined using nonnegative coefficients to obtain n.

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A367222 Number of subsets of {1..n} whose cardinality can be written as a nonnegative linear combination of the elements.

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A367223 Number of subsets of {1..n} whose cardinality cannot be written as a nonnegative linear combination of the elements.

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A326076 Number of subsets of {1..n} containing all of their integer products <= n.

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A326117 Number of subsets of {1..n} containing no products of two or more distinct elements.

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A308546 Number of double-closed subsets of {1..n}.

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A365043 Number of subsets of {1..n} whose greatest element can be written as a (strictly) positive linear combination of the others.

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