A326081 Number of subsets of {1..n} containing the product of any set of distinct elements whose product is <= n.

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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..35.

Formula

For n > 0, a(n) = 2 * A308542(n).

Example

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}

Mathematica

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

Crossrefs

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

Sequence in context: A329824 A229614 A230216 * A245392 A115909 A254940

Adjacent sequences: A326078 A326079 A326080 * A326082 A326083 A326084

Keyword

nonn

Author

Gus Wiseman, Jun 05 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

Status

approved