A326023 -id:A326023 - cp's OEIS frontend

A326083 Number of subsets of {1..n} containing all of their pairwise sums <= n.

1, 2, 3, 5, 7, 12, 16, 27, 37, 58, 80, 131, 171, 277, 380, 580, 785, 1250, 1655, 2616, 3516, 5344, 7257, 11353, 14931, 23204, 31379, 47511, 63778, 98681, 130503, 201357, 270038, 407429, 548090, 840171, 1110429, 1701872, 2284325, 3440337, 4601656

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

The summands are allowed to be equal. The case where they must be distinct is A326080. If A007865 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).
From Gus Wiseman, Jul 08 2019: (Start)
Also the number of subsets of {1..n} containing no sum of any multiset of the elements. For example, the a(0) = 1 through a(6) = 16 subsets are:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {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}    {3,4}
                              {4,5}    {3,5}
                              {3,4,5}  {4,5}
                                       {4,6}
                                       {5,6}
                                       {3,4,5}
                                       {4,5,6}
(End)

			The a(0) = 1 through a(6) = 16 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {2}    {2}      {3}        {3}          {4}
           {1,2}  {3}      {4}        {4}          {5}
                  {2,3}    {2,4}      {5}          {6}
                  {1,2,3}  {3,4}      {2,4}        {3,6}
                           {2,3,4}    {3,4}        {4,5}
                           {1,2,3,4}  {3,5}        {4,6}
                                      {4,5}        {5,6}
                                      {2,4,5}      {2,4,6}
                                      {3,4,5}      {3,4,6}
                                      {2,3,4,5}    {3,5,6}
                                      {1,2,3,4,5}  {4,5,6}
                                                   {2,4,5,6}
                                                   {3,4,5,6}
                                                   {2,3,4,5,6}
                                                   {1,2,3,4,5,6}
The a(7) = 27 subsets:
  {}  {4}  {36}  {246}  {2467}  {24567}  {234567}  {1234567}
      {5}  {45}  {356}  {3467}  {34567}
      {6}  {46}  {367}  {3567}
      {7}  {47}  {456}  {4567}
           {56}  {457}
           {57}  {467}
           {67}  {567}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100

Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326080.

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

For n > 0, a(n) = A103580(n) + 1.

A051026 Number of primitive subsequences of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 17, 33, 45, 73, 103, 205, 253, 505, 733, 1133, 1529, 3057, 3897, 7793, 10241, 16513, 24593, 49185, 59265, 109297, 163369, 262489, 355729, 711457, 879937, 1759873, 2360641, 3908545, 5858113, 10534337, 12701537, 25403073, 38090337, 63299265, 81044097, 162088193, 205482593, 410965185, 570487233, 855676353

Offset: 0

Eric W. Weisstein

a(n) counts all subsequences of {1, ..., n} in which no term divides any other. If n is a prime a(n) = 2*a(n-1)-1 because for each subsequence s counted by a(n-1) two different subsequences are counted by a(n): s and s,n. There is only one exception: 1,n is not a primitive subsequence because 1 divides n. For all n>1: a(n) < 2*a(n-1). - Alois P. Heinz, Mar 07 2011

Maximal primitive subsets are counted by A326077. - Gus Wiseman, Jun 07 2019

			a(4) = 7, the primitive subsequences (including the empty sequence) are: (), (1), (2), (3), (4), (2,3), (3,4).
a(5) = 13 = 2*7-1, the primitive subsequences are: (), (5), (1), (2), (2,5), (3), (3,5), (4), (4,5), (2,3), (2,3,5), (3,4), (3,4,5).
From _Gus Wiseman_, Jun 07 2019: (Start)
The a(0) = 1 through a(5) = 13 primitive (pairwise indivisible) subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {2,3}  {4}    {4}
                       {2,3}  {5}
                       {3,4}  {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {2,3,5}
                              {3,4,5}
a(n) is also the number of subsets of {1..n} containing all of their pairwise products <= n as well as any quotients of divisible elements. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,3}      {1,3}
                  {1,3}    {1,4}      {1,4}
                  {1,2,3}  {1,2,4}    {1,5}
                           {1,3,4}    {1,2,4}
                           {1,2,3,4}  {1,3,4}
                                      {1,3,5}
                                      {1,4,5}
                                      {1,2,3,4}
                                      {1,2,4,5}
                                      {1,3,4,5}
                                      {1,2,3,4,5}
Also the number of subsets of {1..n} containing all of their multiples <= n. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {2}    {2}      {3}        {3}
           {1,2}  {3}      {4}        {4}
                  {2,3}    {2,4}      {5}
                  {1,2,3}  {3,4}      {2,4}
                           {2,3,4}    {3,4}
                           {1,2,3,4}  {3,5}
                                      {4,5}
                                      {2,3,4}
                                      {2,4,5}
                                      {3,4,5}
                                      {2,3,4,5}
                                      {1,2,3,4,5}
(End)
From _Gus Wiseman_, Mar 12 2024: (Start)
Also the number of subsets of {1..n} containing all divisors of the elements. For example, the a(0) = 1 through a(6) = 17 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,2}      {1,2}
                  {1,3}    {1,3}      {1,3}
                  {1,2,3}  {1,2,3}    {1,5}
                           {1,2,4}    {1,2,3}
                           {1,2,3,4}  {1,2,4}
                                      {1,2,5}
                                      {1,3,5}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,2,3,4,5}
(End)

Blanchet-Sadri, Francine. Algorithmic combinatorics on partial words. Chapman & Hall/CRC, Boca Raton, FL, 2008. ii+385 pp. ISBN: 978-1-4200-6092-8; 1-4200-6092-9 MR2384993 (2009f:68142). See p. 320. - N. J. A. Sloane, Apr 06 2012

Juliana Couras, Ricardo Jesus, and Tomás Oliveira e Silva, Table of n, a(n) for n = 0..800 (terms up to n=80 from Alois P. Heinz)
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
Nathan McNew, Counting primitive subsets and other statistics of the divisor graph of {1,2,..,n}, arXiv:1808.04923 [math.NT], 2018.
Richárd Palincza, Counting type and extremal problems from Arithmetic Combinatorics, Ph. D. Thesis, Budapest Univ. Tech. Econ. (Hungary, 2024).
Eric Weisstein's World of Mathematics, Primitive Sequence

Cf. A007865, A054519, A096827, A103580, A303362, A305148.
Cf. A326023, A326076, A326077, A326081, A326082, A326083, A326117.
Cf. A037031, A051026, A355740, A368110, A370585.

with(numtheory):
b:= proc(s) option remember; local n;
      n:= max(s[]);
      `if`(n<0, 1, b(s minus {n}) + b(s minus divisors(n)))
    end:
bb:= n-> b({$2..n} minus divisors(n)):
sb:= proc(n) option remember; `if`(n<2, 0, bb(n) + sb(n-1)) end:
a:= n-> `if`(n=0, 1, `if`(isprime(n), 2*a(n-1)-1, 2+sb(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 07 2011

b[s_] := b[s] = With[{n=Max[s]}, If[n < 0, 1, b[Complement[s, {n}]] + b[Complement[s, Divisors[n]]]]];
bb[n_] := b[Complement[Range[2, n], Divisors[n]]];
sb[n_] := sb[n] = If[n < 2, 0, bb[n] + sb[n-1]];
a[n_] := If[n == 0, 1, If[PrimeQ[n], 2a[n-1] - 1, 2 + sb[n]]]; Table[a[n], {n, 0, 37}]
(* Jean-François Alcover, Jul 27 2011, converted from Maple *)
Table[Length[Select[Subsets[Range[n]], SubsetQ[#,Select[Union@@Table[#*i,{i,n}],#<=n&]]&]],{n,10}] (* Gus Wiseman, Jun 07 2019 *)
Table[Length[Select[Subsets[Range[n]], #==Union@@Divisors/@#&]],{n,0,10}] (* Gus Wiseman, Mar 12 2024 *)

More terms from David Wasserman, May 02 2002

a(32)-a(37) from Donovan Johnson, Aug 11 2010

A326080 Number of subsets of {1..n} containing the sum of every subset whose sum is <= n.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 31, 47, 73, 110, 168, 247, 375, 546, 817, 1193, 1769, 2552, 3791, 5445, 8012, 11517, 16899, 24144, 35391, 50427, 73614, 104984, 152656, 216802, 315689, 447473, 648813, 920163, 1332991, 1884735, 2728020, 3853437, 5568644, 7868096, 11347437

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

Equivalently, a(n) is the number of subsets of {1..n} containing the sum of any two distinct elements whose sum is <= n.
The summands must be distinct. The case where they are allowed to be equal is A326083.
If A151897 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).

			The a(0) = 1 through a(5) = 19 subsets:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {2}    {2}      {2}        {2}
           {1,2}  {3}      {3}        {3}
                  {1,3}    {4}        {4}
                  {2,3}    {1,4}      {5}
                  {1,2,3}  {2,3}      {1,5}
                           {2,4}      {2,4}
                           {3,4}      {2,5}
                           {1,3,4}    {3,4}
                           {2,3,4}    {3,5}
                           {1,2,3,4}  {4,5}
                                      {1,4,5}
                                      {2,3,5}
                                      {2,4,5}
                                      {3,4,5}
                                      {1,3,4,5}
                                      {2,3,4,5}
                                      {1,2,3,4,5}
The a(6) = 31 subsets:
  {}  {1}  {1,6}  {1,5,6}  {1,4,5,6}  {1,3,4,5,6}  {1,2,3,4,5,6}
      {2}  {2,5}  {2,3,5}  {2,3,5,6}  {2,3,4,5,6}
      {3}  {2,6}  {2,4,6}  {2,4,5,6}
      {4}  {3,4}  {2,5,6}  {3,4,5,6}
      {5}  {3,5}  {3,4,5}
      {6}  {3,6}  {3,4,6}
           {4,5}  {3,5,6}
           {4,6}  {4,5,6}
           {5,6}

Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326083.

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

a(n)={
   my(recurse(k, b)=
      if( k > n, 1,
          my(t=self()(k + 1, b + (1<Andrew Howroyd, Aug 30 2019

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

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

A326489 Number of product-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 22, 44, 88, 136, 252, 504, 896, 1792, 3392, 6352, 9720, 19440, 35664, 71328, 129952, 247232, 477664, 955328, 1700416, 2657280, 5184000, 10368000, 19407360, 38814720, 68868352, 137736704, 260693504, 505830400, 999641600, 1882820608, 2807196672

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

A set is product-free if it contains no product of two (not necessarily distinct) elements.

			The a(0) = 1 through a(6) = 22 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}
                             {2,3,5}  {3,5}
                             {3,4,5}  {3,6}
                                      {4,5}
                                      {4,6}
                                      {5,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,4,6}
                                      {3,5,6}
                                      {4,5,6}
                                      {3,4,5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..167, (terms up to a(100) from Andrew Howroyd)
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
Andrew Howroyd, PARI Program

Product-closed subsets are A326076.
Subsets containing no products are A326114.
Subsets containing no products of distinct elements are A326117.
Subsets containing no quotients are A327591.
Maximal product-free subsets are A326496.
Cf. A007865, A051026, A326023, A326081, A326116, A326495.

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

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

a(0)=1 prepended to data, example and b-file by Peter Kagey, Sep 18 2019

A326077 Number of maximal primitive subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 11, 11, 13, 13, 23, 24, 36, 36, 48, 48, 64, 66, 126, 126, 150, 151, 295, 363, 507, 507, 595, 595, 895, 903, 1787, 1788, 2076, 2076, 4132, 4148, 5396, 5396, 6644, 6644, 9740, 11172, 22300, 22300, 26140, 26141, 40733, 40773, 60333, 60333, 80781, 80783

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

a(n) is the number of maximal primitive subsets of {1, ..., n}. Here primitive means that no element of the subset divides any other and maximal means that no element can be added to the subset while maintaining the property of being pairwise indivisible. - Nathan McNew, Aug 10 2020

			The a(0) = 1 through a(9) = 7 sets:
  {}  {1}  {1}  {1}   {1}   {1}    {1}    {1}     {1}     {1}
           {2}  {23}  {23}  {235}  {235}  {2357}  {2357}  {2357}
                      {34}  {345}  {345}  {3457}  {3457}  {2579}
                                   {456}  {4567}  {3578}  {3457}
                                                  {4567}  {3578}
                                                  {5678}  {45679}
                                                          {56789}

Nathan McNew, Table of n, a(n) for n = 0..300

Cf. A001055, A051026 (all primitive subsets), A067992, A096827, A143824, A285572, A285573, A303362, A305148, A305149, A316476, A325861, A326023, A326082.

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],stableQ[#,Divisible]&]]],{n,0,10}]

divset(n)={sumdiv(n, d, if(dif(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 30 2019

Terms a(19) to a(55) from Andrew Howroyd, Aug 30 2019

Name edited by Nathan McNew, Aug 10 2020

A326079 Number of subsets of {1..n} containing all of their integer quotients > 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 48, 96, 144, 288, 432, 864, 1104, 2208, 3312, 5184, 7872, 15744, 20112, 40224, 53376, 84640, 126960, 253920, 309600, 619200, 928800, 1475136, 1984320, 3968640, 4901760, 9803520, 12585600, 20394624, 30591936, 52483392, 65894976, 131789952, 197684928, 323175744, 411685056

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

These sets are closed under taking the quotient of two distinct divisible terms.

			The a(6) = 48 subsets:
  {}  {1}  {1,2}  {1,2,3}  {1,2,3,4}  {1,2,3,4,5}  {1,2,3,4,5,6}
      {2}  {1,3}  {1,2,4}  {1,2,3,5}  {1,2,3,4,6}
      {3}  {1,4}  {1,2,5}  {1,2,3,6}  {1,2,3,5,6}
      {4}  {1,5}  {1,3,4}  {1,2,4,5}  {2,3,4,5,6}
      {5}  {1,6}  {1,3,5}  {1,3,4,5}
      {6}  {2,3}  {1,4,5}  {1,4,5,6}
           {2,4}  {1,4,6}  {2,3,4,5}
           {2,5}  {1,5,6}  {2,3,4,6}
           {3,4}  {2,3,4}  {2,3,5,6}
           {3,5}  {2,3,5}
           {4,5}  {2,3,6}
           {4,6}  {2,4,5}
           {5,6}  {3,4,5}
                  {4,5,6}

Cf. A007865, A051026, A054519, A067992, A103580, A325860, A325994, A326023, A326076, A326078, A326081.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Divide@@@Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]]&]],{n,0,10}]

For n > 0, a(n) = 2 * A326078(n) = 2 * (A326023(n) - 1).

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

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

A326078 Number of subsets of {2..n} containing all of their integer quotients > 1.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 24, 48, 72, 144, 216, 432, 552, 1104, 1656, 2592, 3936, 7872, 10056, 20112, 26688, 42320, 63480, 126960, 154800, 309600, 464400, 737568, 992160, 1984320, 2450880, 4901760, 6292800, 10197312, 15295968, 26241696, 32947488, 65894976, 98842464, 161587872, 205842528

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

These sets are closed under taking the quotient of two distinct divisible terms.

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

Cf. A007865, A051026, A054519, A067992, A103580, A325860, A325994, A326023, A326076, A326079, A326081.

Table[Length[Select[Subsets[Range[2,n]],SubsetQ[#,Divide@@@Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]]&]],{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) + if(accept(b, k), self()(k, b + (1<Andrew Howroyd, Aug 30 2019

For n > 0, a(n) = A326023(n) - 1.

For n > 0, a(n) = A326079(n)/2.

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

cp's OEIS Frontend

A326083 Number of subsets of {1..n} containing all of their pairwise sums <= n.

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A051026 Number of primitive subsequences of {1, 2, ..., n}.

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A326080 Number of subsets of {1..n} containing the sum of every subset whose sum is <= n.

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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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A326489 Number of product-free subsets of {1..n}.

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A326077 Number of maximal primitive subsets of {1..n}.

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