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

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

Author

Gus Wiseman, Jun 05 2019

Keywords

• nonn

Comments

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

Examples

			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}
		

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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

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

Extensions

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