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

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

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) = 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}
		

Crossrefs

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

Programs

Mathematica

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

Formula

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

Extensions

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