A365046 -id:A365046 - cp's OEIS frontend

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

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

A365045 Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.

Original entry on oeis.org

0, 1, 1, 2, 4, 11, 23, 53, 111, 235, 483, 988, 1998, 4036, 8114, 16289, 32645, 65389, 130887, 261923, 524014, 1048251, 2096753, 4193832, 8388034, 16776544, 33553622, 67107919, 134216597, 268434140, 536869355, 1073740012, 2147481511, 4294964834, 8589931700

Offset: 0

Gus Wiseman, Aug 24 2023

nonn

Also subsets of {1..n} containing n whose greatest element cannot be written as a positive linear combination of the others.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is not counted under a(10).
The a(0) = 0 through a(5) = 11 subsets:
  .  {1}  {2}  {3}    {4}        {5}
               {2,3}  {3,4}      {2,5}
                      {2,3,4}    {3,5}
                      {1,2,3,4}  {4,5}
                                 {2,4,5}
                                 {3,4,5}
                                 {1,2,3,5}
                                 {1,2,4,5}
                                 {1,3,4,5}
                                 {2,3,4,5}
                                 {1,2,3,4,5}

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

The nonempty case is A070880.
The nonnegative version is A124506, first differences of A326083.
The binary version is A288728, first differences of A007865.
A subclass is A341507.
The complement is counted by A365042, first differences of A365043.
First differences of A365044.
The nonnegative complement is A365046, first differences of A364914.
The binary complement is A365070, first differences of A093971.
Without re-usable parts we have A365071, first differences of A151897.
A085489 and A364755 count subsets w/o the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364534, A365069.

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]&&And@@Table[combp[#[[k]],Union[Delete[#,k]]]=={},{k,Length[#]}]&]],{n,0,10}]

a(n) = A070880(n) + 1 for n > 0.

A365314 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 1, 3, 6, 8, 14, 14, 23, 24, 33, 28, 52, 36, 55, 58, 73, 53, 95, 62, 110, 94, 105, 81, 165, 105, 133, 132, 176, 112, 225, 123, 210, 174, 192, 186, 306, 157, 223, 218, 328, 180, 354, 192, 324, 315, 288, 216, 474, 260, 383, 311, 404, 254, 491, 338, 511, 360

Offset: 0

Gus Wiseman, Sep 05 2023

nonn

Is there only one case of nonzero adjacent equal parts, at position n = 6?

			We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
The a(2) = 1 through a(7) = 14 pairs:
  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)
         (1,3)  (1,3)  (1,3)  (1,3)  (1,3)
         (2,3)  (1,4)  (1,4)  (1,4)  (1,4)
                (2,3)  (1,5)  (1,5)  (1,5)
                (2,4)  (2,3)  (1,6)  (1,6)
                (3,4)  (2,5)  (2,3)  (1,7)
                       (3,5)  (2,4)  (2,3)
                       (4,5)  (2,5)  (2,5)
                              (2,6)  (2,7)
                              (3,4)  (3,4)
                              (3,5)  (3,7)
                              (3,6)  (4,7)
                              (4,6)  (5,7)
                              (5,6)  (6,7)

Chai Wah Wu, Table of n, a(n) for n = 0..10000

The unrestricted version is A000217, ranks A001358.
For all subsets instead of just pairs we have A365073, complement A365380.
For strict partitions we have A365311, complement A365312.
The case of positive coefficients is A365315, for all subsets A088314.
The binary complement is A365320, positive A365321.
For partitions we have A365379, complement A365378.
A004526 counts partitions of length 2, shift right for strict.
A007865 counts sum-free subsets, complement A093971.
A179822 and A326080 count sum-closed subsets.
A364350 counts combination-free strict partitions.
A364914/A365046 count combination-full subsets, complement A326083/A124506.
Cf. A070880, A088809, A151897, A326020, A365043, A365322, A365383.

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 A365314(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 len(a) # Chai Wah Wu, Sep 12 2023

A365321 Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 10, 13, 18, 24, 30, 37, 46, 54, 63, 77, 85, 99, 111, 127, 141, 161, 171, 194, 210, 235, 246, 277, 293, 322, 342, 372, 389, 428, 441, 491, 504, 545, 561, 612, 635, 680, 701, 753, 773, 836, 846, 911, 932, 1000, 1017, 1082, 1103, 1176, 1193

Offset: 0

Gus Wiseman, Sep 06 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.

			For the pair p = (2,3) we have 4 = 2*2 + 0*3, so p is not counted under A365320(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is counted under a(4).
The a(2) = 1 through a(7) = 13 pairs:
  (1,2)  (1,3)  (1,4)  (1,5)  (1,6)  (1,7)
         (2,3)  (2,3)  (2,4)  (2,3)  (2,4)
                (2,4)  (2,5)  (2,5)  (2,6)
                (3,4)  (3,4)  (2,6)  (2,7)
                       (3,5)  (3,4)  (3,5)
                       (4,5)  (3,5)  (3,6)
                              (3,6)  (3,7)
                              (4,5)  (4,5)
                              (4,6)  (4,6)
                              (5,6)  (4,7)
                                     (5,6)
                                     (5,7)
                                     (6,7)

The unrestricted version is A000217, ranks A001358.
For strict partitions we have A088528, complement A088314.
The (binary) complement is A365315, nonnegative A365314.
For nonnegative coefficients we have A365320, for subsets A365380.
For all subsets instead of just pairs we have A365322, complement A088314.
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 count combination-free subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A070880, A088571, A088809, A151897, A326020, A365043, A365073, A365311, A365312, A365378, A365380.

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],{2}], combp[n,#]=={}&]],{n,0,30}]

from itertools import count
from sympy import divisors
def A365321(n):
    a = set()
    for i in range(1,n+1):
        for j in count(i,i):
            if j >= n:
                break
            for d in divisors(n-j):
                if d>=i:
                    break
                a.add((d,i))
    return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 12 2023

A365044 Number of subsets of {1..n} whose greatest element cannot be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

1, 2, 3, 5, 9, 20, 43, 96, 207, 442, 925, 1913, 3911, 7947, 16061, 32350, 64995, 130384, 261271, 523194, 1047208, 2095459, 4192212, 8386044, 16774078, 33550622, 67104244, 134212163, 268428760, 536862900, 1073732255, 2147472267, 4294953778, 8589918612, 17179850312

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

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

Also subsets of {1..n} such that no element can be written as a (strictly) positive linear combination of the others.

			The subset S = {3,5,6,8} has 6 = 2*3 + 0*5 + 0*8 and 8 = 1*3 + 1*5 + 0*6 but neither of these is strictly positive, so S is counted under a(8).
The a(0) = 1 through a(5) = 20 subsets:
  {}  {}   {}   {}     {}         {}
      {1}  {1}  {1}    {1}        {1}
           {2}  {2}    {2}        {2}
                {3}    {3}        {3}
                {2,3}  {4}        {4}
                       {2,3}      {5}
                       {3,4}      {2,3}
                       {2,3,4}    {2,5}
                       {1,2,3,4}  {3,4}
                                  {3,5}
                                  {4,5}
                                  {2,3,4}
                                  {2,4,5}
                                  {3,4,5}
                                  {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}

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

The binary version is A007865, first differences A288728.
The binary complement is A093971, first differences A365070.
Without re-usable parts we have A151897, first differences A365071.
The nonnegative version is A326083, first differences A124506.
A subclass is A341507.
The nonnegative complement is A364914, first differences A365046.
The complement is counted by A365043, first differences A365042.
First differences are A365045.
A085489 and A364755 count subsets w/o the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A006951, A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364534, A365069.

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]],And@@Table[combp[Last[#],Union[Most[#]]]=={},{k,Length[#]}]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A365044(n):
    mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
    return n+1+sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] not in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023

a(n) = 2^n - A365043(n).

a(15)-a(34) from Chai Wah Wu, Nov 20 2023

A365315 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using positive coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 5, 8, 10, 12, 15, 18, 20, 24, 28, 28, 35, 37, 42, 44, 49, 49, 60, 59, 66, 65, 79, 74, 85, 84, 93, 93, 107, 100, 120, 104, 126, 121, 142, 129, 145, 140, 160, 150, 173, 154, 189, 170, 196, 176, 208, 193, 223, 202, 238, 203, 241, 227, 267, 235

Offset: 0

Gus Wiseman, Sep 06 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.

			We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
For the pair p = (2,3), we have 4 = 2*2 + 0*3, so p is counted under A365314(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is not counted under a(4).
The a(3) = 1 through a(10) = 15 pairs:
  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)
         (1,3)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)
                (1,4)  (1,4)  (1,4)  (1,4)  (1,4)  (1,4)
                (2,3)  (1,5)  (1,5)  (1,5)  (1,5)  (1,5)
                       (2,4)  (1,6)  (1,6)  (1,6)  (1,6)
                              (2,3)  (1,7)  (1,7)  (1,7)
                              (2,5)  (2,3)  (1,8)  (1,8)
                              (3,4)  (2,4)  (2,3)  (1,9)
                                     (2,6)  (2,5)  (2,3)
                                     (3,5)  (2,7)  (2,4)
                                            (3,6)  (2,6)
                                            (4,5)  (2,8)
                                                   (3,4)
                                                   (3,7)
                                                   (4,6)

Chai Wah Wu, Table of n, a(n) for n = 0..10000

The unrestricted version is A000217, ranks A001358.
For all subsets instead of just pairs we have A088314, complement A365322.
For strict partitions we have A088571, complement A088528.
The case of nonnegative coefficients is A365314, for all subsets A365073.
The (binary) complement is A365321, nonnegative A365320.
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, A088809, A326020, A364534, A365043, A365311, A365312, A365378, A365379, A365380, A365383.

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],{2}],combp[n,#]!={}&]],{n,0,30}]

from itertools import count
from sympy import divisors
def A365315(n):
    a = set()
    for i in range(1,n+1):
        for j in count(i,i):
            if j >= n:
                break
            for d in divisors(n-j):
                if d>=i:
                    break
                a.add((d,i))
    return len(a) # Chai Wah Wu, Sep 13 2023

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

Original entry on oeis.org

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.

A365070 Number of subsets of {1..n} containing n and some element equal to the sum of two other (possibly equal) elements.

Original entry on oeis.org

0, 0, 1, 1, 5, 9, 24, 46, 109, 209, 469, 922, 1932, 3858, 7952, 15831, 32214, 64351, 129813, 259566, 521681, 1042703, 2091626, 4182470, 8376007, 16752524, 33530042, 67055129, 134165194, 268328011, 536763582, 1073523097, 2147268041, 4294505929, 8589506814, 17178978145

Offset: 0

Gus Wiseman, Aug 24 2023

nonn

These are binary sum-full sets where elements can be re-used. The complement is counted by A288728. The non-binary version is A365046, complement A124506. For non-re-usable parts we have A364756, complement A085489.

			The subset {1,3} has no element equal to the sum of two others, so is not counted under a(3).
The subset {3,4,5} has no element equal to the sum of two others, so is not counted under a(5).
The subset {1,3,4} has 4 = 1 + 3, so is counted under a(4).
The subset {2,4,5} has 4 = 2 + 2, so is counted under a(5).
The a(0) = 0 through a(5) = 9 subsets:
  .  .  {1,2}  {1,2,3}  {2,4}      {1,2,5}
                        {1,2,4}    {1,4,5}
                        {1,3,4}    {2,3,5}
                        {2,3,4}    {2,4,5}
                        {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}

Andrew Howroyd, Table of n, a(n) for n = 0..88
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The complement w/o re-usable parts is A085489, first differences of A364755.
First differences of A093971.
The non-binary complement is A124506, first differences of A326083.
The complement is counted by A288728, first differences of A007865.
For partitions (not requiring n) we have A363225, strict A363226.
The case without re-usable parts is A364756, firsts differences of A088809.
The non-binary version is A365046, first differences of A364914.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
A365006 counts no positive combination-full strict ptns.
Cf. A050291, A051026, A151897, A236912, A237113, A237668, A326080, A364349, A364533, A364670.

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

First differences of A093971.

a(21) onwards added (using A093971) by Andrew Howroyd, Jan 13 2024

A365069 Number of subsets of {1..n} containing n and some element equal to the sum of two or more distinct other elements. A variation of non-binary sum-full subsets without re-usable elements.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 17, 41, 88, 201, 418, 892, 1838, 3798, 7716, 15740

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

The complement is counted by A365071. The binary case is A364756. Allowing elements to be re-used gives A365070. A version for partitions (but not requiring n) is A237668.

			The subset {2,4,6} has 6 = 4 + 2 so is counted under a(6).
The subset {1,2,4,7} has 7 = 4 + 2 + 1 so is counted under a(7).
The subset {1,4,5,8} has 5 = 4 + 1 so is counted under a(8).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
                    {1,2,3,4}  {2,3,5}      {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}
                                            {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}

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

The complement w/ re-usable parts is A288728, first differences of A007865.
First differences of A364534.
The binary complement is A364755, first differences of A085489.
The binary version is A364756, first differences of A088809.
The version with re-usable parts is A365070, first differences of A093971.
The complement is counted by A365071, first differences of A151897.
A124506 counts nonnegative combination-free subsets, differences of A326083.
A365046 counts nonnegative combination-full subsets, differences of A364914.
For partitions: A108917, A236912, A237113, A237668, A364532, A364913.
Strict partitions: A116861, A364272, A364349, A364350, A364839, A364916.
Cf. A050291, A326080, A363226, A364346, A364348, A364670.

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

a(n) = 2^(n-1) - A365070(n).

First differences of A364534.

A365659 Number of strict integer partitions of n that either have (1) length 2, or (2) greatest part n/2.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 4, 4, 6, 5, 8, 6, 10, 7, 12, 8, 15, 9, 18, 10, 21, 11, 25, 12, 29, 13, 34, 14, 40, 15, 46, 16, 53, 17, 62, 18, 71, 19, 82, 20, 95, 21, 109, 22, 125, 23, 144, 24, 165, 25, 189, 26, 217, 27, 248, 28, 283, 29, 324

Offset: 0

Gus Wiseman, Sep 16 2023

nonn

Also the number of strict integer partitions of n containing two possibly equal elements summing to n.

			The a(3) = 1 through a(11) = 5 partitions:
  (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)    (5,4)  (6,4)    (6,5)
                (4,1)  (5,1)    (5,2)  (6,2)    (6,3)  (7,3)    (7,4)
                       (3,2,1)  (6,1)  (7,1)    (7,2)  (8,2)    (8,3)
                                       (4,3,1)  (8,1)  (9,1)    (9,2)
                                                       (5,3,2)  (10,1)
                                                       (5,4,1)

Without repeated parts we have A140106.
The non-strict version is A238628.
For subsets instead of strict partitions we have A365544.
A000009 counts subsets summing to n.
A365046 counts combination-full subsets, differences of A364914.
A365543 counts partitions of n with a submultiset summing to k.
Cf. A008967, A046663, A068911, A095944, A364272, A365376, A365377.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(Length[#]==2||Max@@#==n/2)&]], {n,0,30}]

from sympy.utilities.iterables import partitions
def A365659(n): return n>>1 if n&1 or n==0 else (m:=n>>1)+sum(1 for p in partitions(m) if max(p.values(),default=1)==1)-2 # Chai Wah Wu, Sep 18 2023

a(n) = (n-1)/2 if n is odd. a(n) = n/2 + A000009(n/2) - 2 if n is even and n > 0. - Chai Wah Wu, Sep 18 2023

cp's OEIS Frontend

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

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A365045 Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.

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A365314 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.

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A365321 Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.

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A365044 Number of subsets of {1..n} whose greatest element cannot be written as a (strictly) positive linear combination of the others.

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A365315 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using positive coefficients to obtain n.

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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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