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Number of subsets of {1..n} containing all of their integer quotients > 1.
11

%I #8 Aug 30 2019 21:47:29

%S 1,2,4,8,16,32,48,96,144,288,432,864,1104,2208,3312,5184,7872,15744,

%T 20112,40224,53376,84640,126960,253920,309600,619200,928800,1475136,

%U 1984320,3968640,4901760,9803520,12585600,20394624,30591936,52483392,65894976,131789952,197684928,323175744,411685056

%N Number of subsets of {1..n} containing all of their integer quotients > 1.

%C These sets are closed under taking the quotient of two distinct divisible terms.

%F For n > 0, a(n) = 2 * A326078(n) = 2 * (A326023(n) - 1).

%e The a(6) = 48 subsets:

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

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

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

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

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

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

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

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

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

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

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

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

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

%e {4,5,6}

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

%Y Cf. A007865, A051026, A054519, A067992, A103580, A325860, A325994, A326023, A326076, A326078, A326081.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jun 05 2019

%E Terms a(21) and beyond from _Andrew Howroyd_, Aug 30 2019