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A326081
Number of subsets of {1..n} containing the product of any set of distinct elements whose product is <= n.
9
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.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 05 2019
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019
STATUS
approved