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Number of subsets of {1..n} containing all differences and quotients of pairs of distinct elements.
2

%I #8 Aug 25 2019 19:45:12

%S 1,2,4,6,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,

%T 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,

%U 95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127

%N Number of subsets of {1..n} containing all differences and quotients of pairs of distinct elements.

%C 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

%F a(n) = 2*n + 1 = A005408(n) for n > 3. - _Andrew Howroyd_, Aug 25 2019

%e The a(0) = 1 through a(6) = 13 subsets:

%e {} {} {} {} {} {} {}

%e {1} {1} {1} {1} {1} {1}

%e {2} {2} {2} {2} {2}

%e {1,2} {3} {3} {3} {3}

%e {1,2} {4} {4} {4}

%e {1,2,3} {1,2} {5} {5}

%e {2,4} {1,2} {6}

%e {1,2,3} {2,4} {1,2}

%e {1,2,3,4} {1,2,3} {2,4}

%e {1,2,3,4} {1,2,3}

%e {1,2,3,4,5} {1,2,3,4}

%e {1,2,3,4,5}

%e {1,2,3,4,5,6}

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

%Y Subsets with difference are A054519.

%Y Subsets with quotients are A326023.

%Y Subsets with quotients > 1 are A326079.

%Y Subsets without differences or quotients are A326490.

%Y Cf. A005408, A007865, A051026, A325849, A326076, A326491.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jul 09 2019

%E Terms a(20) and beyond from _Andrew Howroyd_, Aug 25 2019