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

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

Author

Gus Wiseman, Jun 06 2019

Keywords

• nonn

Comments

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

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

Examples

			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}
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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

Extensions

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