A326020 -id:A326020 - cp's OEIS frontend

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

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

A070880 Consider the 2^(n-1)-1 nonempty subsets S of {1, 2, ..., n-1}; a(n) gives number of such S for which it is impossible to partition n into parts from S such that each s in S is used at least once.

Original entry on oeis.org

0, 0, 1, 3, 10, 22, 52, 110, 234, 482, 987, 1997, 4035, 8113, 16288, 32644, 65388, 130886, 261922, 524013, 1048250, 2096752, 4193831, 8388033, 16776543, 33553621, 67107918, 134216596, 268434139, 536869354, 1073740011, 2147481510, 4294964833, 8589931699

Offset: 1

Naohiro Nomoto, Nov 16 2003

Also the number of nonempty subsets of {1..n-1} that cannot be linearly combined using all positive coefficients to obtain n. - Gus Wiseman, Sep 10 2023

			a(4)=3 because there are three different subsets S of {1,2,3} satisfying the condition: {3}, {2,3} & {1,2,3}. For the other subsets S, such as {1,2}, there is a partition of 4 which uses them all (such as 4 = 1+1+2).
From _Gus Wiseman_, Sep 10 2023: (Start)
The a(6) = 22 subsets:
  {4}  {2,3}  {1,2,4}  {1,2,3,4}  {1,2,3,4,5}
  {5}  {2,5}  {1,2,5}  {1,2,3,5}
       {3,4}  {1,3,4}  {1,2,4,5}
       {3,5}  {1,3,5}  {1,3,4,5}
       {4,5}  {1,4,5}  {2,3,4,5}
              {2,3,4}
              {2,3,5}
              {2,4,5}
              {3,4,5}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..100

For sets with sum < n instead of maximum < n we have A088528.
The complement is counted by A365042, including empty set A088314.
Allowing empty sets gives A365045, nonnegative version apparently A124506.
Without re-usable parts we have A365377(n) - 1.
For nonnegative (instead of positive) coefficients we have A365380(n) - 1.
A326083 counts combination-free subsets, complement A364914.
A364350 counts combination-free strict partitions, complement A364913.
Cf. A007865, A085489, A093971, A151897, A326020, A326080, A364534, A365044.

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-1]]], combp[n,#]=={}&]],{n,7}] (* Gus Wiseman, Sep 10 2023 *)

from sympy.utilities.iterables import partitions
def A070880(n): return (1<Chai Wah Wu, Sep 10 2023

a(n) = 2^(n-1) - A088314(n). - Charlie Neder, Feb 08 2019

a(n) = A365045(n) - 1. - Gus Wiseman, Sep 10 2023

Edited by N. J. A. Sloane, Sep 09 2017

a(20)-a(34) from Alois P. Heinz, Feb 08 2019

A121269 Number of maximal sum-free subsets of {1,2,...,n}.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 8, 13, 17, 23, 29, 37, 51, 66, 86, 118, 158, 201, 265, 359, 471, 598, 797, 1043, 1378, 1765, 2311, 3064, 3970, 5017, 6537, 8547, 11020, 14007, 18026, 23404, 30026, 37989, 48945, 62759, 80256, 101070, 129193, 164835, 209279, 262693, 334127

Offset: 0

N. Hindman (nhindman(AT)aol.com), Aug 23 2006

nonn

Also the number of maximal subsets of {1..n} containing no differences of pairs of elements. - Gus Wiseman, Jul 10 2019

			a(5)=5 because the maximal sum-free subsets of {1,2,3,4,5} are {1,4}, {2,3}, {2,5}, {1,3,5} and {3,4,5}
From _Gus Wiseman_, Jul 10 2019: (Start)
The a(1) = 1 through a(8) = 13 subsets:
  {1}  {1}  {1,3}  {1,3}  {1,4}    {2,3}    {1,4,6}    {1,3,8}
       {2}  {2,3}  {1,4}  {2,3}    {1,3,5}  {1,4,7}    {1,4,6}
                   {2,3}  {2,5}    {1,4,6}  {2,3,7}    {1,4,7}
                   {3,4}  {1,3,5}  {2,5,6}  {2,5,6}    {1,5,8}
                          {3,4,5}  {3,4,5}  {2,6,7}    {1,6,8}
                                   {4,5,6}  {3,4,5}    {2,5,6}
                                            {1,3,5,7}  {2,5,8}
                                            {4,5,6,7}  {2,6,7}
                                                       {3,4,5}
                                                       {1,3,5,7}
                                                       {2,3,7,8}
                                                       {4,5,6,7}
                                                       {5,6,7,8}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..80
P. J. Cameron and P. Erdős, On the number of integers with various properties, in R. A. Mullin, ed., Number Theory: Proc. First Conf. of Canad. Number Theory Assoc. Conf., Banff, De Gruyter, Berlin, 1990, pp. 61-79.
N. Hindman and H. Jordan, Measures of sum-free intersecting families, New York J. Math. 13 (2007), 97-106.

Maximal product-free subsets are A326496.
Sum-free subsets are A007865.
Maximal sum-free and product-free subsets are A326497.
Subsets with sums are A326083.
Maximal subsets without sums of distinct elements are A326498.
Cf. A103580, A326020, A326489, A326495.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Plus@@@Tuples[#,2]]=={}&]]],{n,0,10}] (* Gus Wiseman, Jul 10 2019 *)

a(0) = 1 prepended by Gus Wiseman, Jul 10 2019

Terms a(42) and beyond from Fausto A. C. Cariboni, Oct 26 2020

A326081 Number of subsets of {1..n} containing the product of any set of distinct elements whose product is <= n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 56, 112, 200, 400, 728, 1456, 2368, 4736, 8896, 16112, 30016, 60032, 105472, 210944, 366848, 679680, 1327232, 2654464, 4434176, 8868352, 17488640, 33118336, 60069248, 120138496, 206804224, 413608448, 759882880, 1461600128, 2909298496, 5319739328

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

For n > 0, this sequence divided by 2 first differs from A326116 at a(12)/2 = 1184, A326116(12) = 1232.
If A326117 counts product-free sets, this sequence counts product-closed sets.
The non-strict case is A326076.

			The a(6) = 56 subsets:
  {}  {1}  {1,2}  {1,2,4}  {1,2,3,6}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {2}  {1,3}  {1,2,5}  {1,2,4,5}  {1,2,3,5,6}
      {3}  {1,4}  {1,2,6}  {1,2,4,6}  {1,2,4,5,6}
      {4}  {1,5}  {1,3,4}  {1,2,5,6}  {1,3,4,5,6}
      {5}  {1,6}  {1,3,5}  {1,3,4,5}  {2,3,4,5,6}
      {6}  {2,4}  {1,3,6}  {1,3,4,6}
           {2,5}  {1,4,5}  {1,3,5,6}
           {2,6}  {1,4,6}  {1,4,5,6}
           {3,4}  {1,5,6}  {2,3,4,6}
           {3,5}  {2,3,6}  {2,3,5,6}
           {3,6}  {2,4,5}  {2,4,5,6}
           {4,5}  {2,4,6}  {3,4,5,6}
           {4,6}  {2,5,6}
           {5,6}  {3,4,5}
                  {3,4,6}
                  {3,5,6}
                  {4,5,6}

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

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

For n > 0, a(n) = 2 * A308542(n).

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

A326495 Number of subsets of {1..n} containing no sums or products of pairs of elements.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 30, 45, 71, 101, 171, 258, 427, 606, 988, 1328, 2141, 3116, 4952, 6955, 11031, 15320, 23978, 33379, 48698, 66848, 104852, 144711, 220757, 304132, 461579, 636555, 973842, 1316512, 1958827, 2585432, 3882842, 5237092, 7884276, 10555738, 15729292

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

The pairs are not required to be strict.

			The a(1) = 1 through a(6) = 17 subsets:
  {}  {}   {}     {}     {}       {}
      {2}  {2}    {2}    {2}      {2}
           {3}    {3}    {3}      {3}
           {2,3}  {4}    {4}      {4}
                  {2,3}  {5}      {5}
                  {3,4}  {2,3}    {6}
                         {2,5}    {2,3}
                         {3,4}    {2,5}
                         {3,5}    {2,6}
                         {4,5}    {3,4}
                         {3,4,5}  {3,5}
                                  {4,5}
                                  {4,6}
                                  {5,6}
                                  {2,5,6}
                                  {3,4,5}
                                  {4,5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..90

Subsets without sums are A007865.
Subsets without products are A326489.
Subsets without differences or quotients are A326490.
Maximal subsets without sums or products are A326497.
Subsets with sums (and products) are A326083.
Cf. A051026, A103580, A326020, A326076, A326117, A326491.

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

For n > 0, a(n) = A326490(n) - 1.

a(19)-a(41) from Andrew Howroyd, Aug 25 2019

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

A326116 Number of subsets of {2..n} containing no products of two or more distinct elements.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 56, 100, 200, 364, 728, 1232, 2464, 4592, 8296, 15920, 31840, 55952, 111904, 195712, 362336, 697360, 1394720, 2334112, 4668224, 9095392, 17225312, 31242784, 62485568, 106668608, 213337216, 392606528, 755131840, 1491146912, 2727555424, 4947175712

Offset: 1

Gus Wiseman, Jun 06 2019

nonn

First differs from A308542 at a(12) = 1232, A308542(12) = 1184.

If this sequence counts product-free sets, A308542 counts product-closed sets.

			The a(6) = 28 subsets:
  {}  {2}  {2,3}  {2,3,4}  {2,3,4,5}
      {3}  {2,4}  {2,3,5}  {2,4,5,6}
      {4}  {2,5}  {2,4,5}  {3,4,5,6}
      {5}  {2,6}  {2,4,6}
      {6}  {3,4}  {2,5,6}
           {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 = 1..47
P. J. Cameron and P. Erdős, On the number of integers with various properties, in R. A. Mullin, ed., Number Theory: Proc. First Conf. of Canad. Number Theory Assoc. Conf., Banff, De Gruyter, Berlin, 1990, pp. 61-79.

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326117, A308542.

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

a(n)={
   my(recurse(k, ep)=
    if(k > n, 1,
      my(t = self()(k + 1, ep));
      if(!bittest(ep,k),
         forstep(i=n\k, 1, -1, if(bittest(ep,i), ep=bitor(ep,1<<(k*i))));
         t += self()(k + 1, ep);
      );
      t);
   );
   recurse(2, 2);
} \\ Andrew Howroyd, Aug 25 2019

For n > 0, a(n) = A326117(n) - 1.

Terms a(21)-a(36) from Andrew Howroyd, Aug 25 2019

cp's OEIS Frontend

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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A070880 Consider the 2^(n-1)-1 nonempty subsets S of {1, 2, ..., n-1}; a(n) gives number of such S for which it is impossible to partition n into parts from S such that each s in S is used at least once.

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A121269 Number of maximal sum-free subsets of {1,2,...,n}.

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A326081 Number of subsets of {1..n} containing the product of any set of distinct elements whose product is <= n.

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A326495 Number of subsets of {1..n} containing no sums or products of pairs of elements.

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