OFFSET
0,2
COMMENTS
The only allowed sets are the empty set, any singleton, any initial interval of positive integers and {2,4}. This can be shown by induction. - Andrew Howroyd, Aug 25 2019
FORMULA
a(n) = 2*n + 1 = A005408(n) for n > 3. - Andrew Howroyd, Aug 25 2019
EXAMPLE
The a(0) = 1 through a(6) = 13 subsets:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{1,2} {3} {3} {3} {3}
{1,2} {4} {4} {4}
{1,2,3} {1,2} {5} {5}
{2,4} {1,2} {6}
{1,2,3} {2,4} {1,2}
{1,2,3,4} {1,2,3} {2,4}
{1,2,3,4} {1,2,3}
{1,2,3,4,5} {1,2,3,4}
{1,2,3,4,5}
{1,2,3,4,5,6}
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], SubsetQ[#, Union[Divide@@@Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&], Subtract@@@Select[Tuples[#, 2], Greater@@#&]]]&]], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 09 2019
EXTENSIONS
Terms a(20) and beyond from Andrew Howroyd, Aug 25 2019
STATUS
approved