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A326494
Number of subsets of {1..n} containing all differences and quotients of pairs of distinct elements.
2
1, 2, 4, 6, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127
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
Subsets with difference are A054519.
Subsets with quotients are A326023.
Subsets with quotients > 1 are A326079.
Subsets without differences or quotients are A326490.
Sequence in context: A024978 A191194 A191231 * A285593 A351631 A248898
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 09 2019
EXTENSIONS
Terms a(20) and beyond from Andrew Howroyd, Aug 25 2019
STATUS
approved