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Number of subsets of {2..n} containing all of their integer quotients > 1.
8

%I #11 Aug 30 2019 21:47:06

%S 1,1,2,4,8,16,24,48,72,144,216,432,552,1104,1656,2592,3936,7872,10056,

%T 20112,26688,42320,63480,126960,154800,309600,464400,737568,992160,

%U 1984320,2450880,4901760,6292800,10197312,15295968,26241696,32947488,65894976,98842464,161587872,205842528

%N Number of subsets of {2..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) = A326023(n) - 1.

%F For n > 0, a(n) = A326079(n)/2.

%e The a(6) = 24 subsets:

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

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

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

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

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

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

%e {4,6}

%e {5,6}

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

%o (PARI)

%o a(n)={

%o my(lim=vector(n, k, sqrtint(k)));

%o my(accept(b, k)=for(i=2, lim[k], if(k%i ==0 && bittest(b,i) != bittest(b,k/i), return(0))); 1);

%o my(recurse(k, b)=

%o my(m=1);

%o for(j=max(2*k,n\2+1), min(2*k+1,n), if(accept(b,j), m*=2));

%o k++;

%o m*if(k > n\2, 1, (self()(k, b) + if(accept(b, k), self()(k, b + (1<<k)))))

%o );

%o recurse(1, 0);

%o } \\ _Andrew Howroyd_, Aug 30 2019

%Y Cf. A007865, A051026, A054519, A067992, A103580, A325860, A325994, A326023, A326076, A326079, A326081.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jun 05 2019

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