A326114 -id:A326114 - cp's OEIS frontend

A103580 Number of nonempty subsets S of {1,2,3,...,n} that have the property that no element x of S is a nonnegative integer linear combination of elements of S-{x}.

1, 2, 4, 6, 11, 15, 26, 36, 57, 79, 130, 170, 276, 379, 579, 784, 1249, 1654, 2615, 3515, 5343, 7256, 11352, 14930, 23203, 31378, 47510, 63777, 98680, 130502, 201356, 270037, 407428, 548089, 840170, 1110428, 1701871, 2284324, 3440336, 4601655

Offset: 1

Jeffrey Shallit, Mar 23 2005

nonn

			a(4) = 6 because the only permissible subsets are {1}, {2}, {3}, {4}, {2,3}, {3,4}.
From _Gus Wiseman_, Jun 07 2019: (Start)
The a(1) = 1 through a(6) = 15 nonempty subsets of {1..n} containing none of their own non-singleton nonzero nonnegative linear combinations 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}
a(n) is also the number of nonempty subsets of {1..n} containing all of their own nonzero nonnegative linear combinations <= n. For example the a(1) = 1 through a(6) = 15 subsets are:
  {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}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..100
Sergey Kitaev, Independent Sets on Path-Schemes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.2.
Sean Li, Counting numerical semigroups by Frobenius number, multiplicity, and depth, arXiv:2208.14587 [math.CO], 2022.

Cf. A007865, A050291, A051026, A085489, A139384, A151897, A308546.

Cf. A326020, A326076, A326080, A326083, A326114.

Table[Length[Select[Subsets[Range[n],{1,n}],SubsetQ[#,Select[Plus@@@Tuples[#,2],#<=n&]]&]],{n,10}] (* Gus Wiseman, Jun 07 2019 *)

a(n) = A326083(n) - 1. - Gus Wiseman, Jun 07 2019

More terms from David Wasserman, Apr 16 2008

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

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

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

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Nov 13 2022

nonn

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

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

Inverse Moebius transform of A103580.

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

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

cp's OEIS Frontend

A103580 Number of nonempty subsets S of {1,2,3,...,n} that have the property that no element x of S is a nonnegative integer linear combination of elements of S-{x}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula