cp's OEIS Frontend

A326078 Number of subsets of {2..n} containing all of their integer quotients > 1.

%I A326078 #11 Aug 30 2019 21:47:06
%S A326078 1,1,2,4,8,16,24,48,72,144,216,432,552,1104,1656,2592,3936,7872,10056,
%T A326078 20112,26688,42320,63480,126960,154800,309600,464400,737568,992160,
%U A326078 1984320,2450880,4901760,6292800,10197312,15295968,26241696,32947488,65894976,98842464,161587872,205842528
%N A326078 Number of subsets of {2..n} containing all of their integer quotients > 1.
%C A326078 These sets are closed under taking the quotient of two distinct divisible terms.
%F A326078 For n > 0, a(n) = A326023(n) - 1.
%F A326078 For n > 0, a(n) = A326079(n)/2.
%e A326078 The a(6) = 24 subsets:
%e A326078   {}  {2}  {2,3}  {2,3,4}  {2,3,4,5}  {2,3,4,5,6}
%e A326078       {3}  {2,4}  {2,3,5}  {2,3,4,6}
%e A326078       {4}  {2,5}  {2,3,6}  {2,3,5,6}
%e A326078       {5}  {3,4}  {2,4,5}
%e A326078       {6}  {3,5}  {3,4,5}
%e A326078            {4,5}  {4,5,6}
%e A326078            {4,6}
%e A326078            {5,6}
%t A326078 Table[Length[Select[Subsets[Range[2,n]],SubsetQ[#,Divide@@@Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]]&]],{n,0,10}]
%o A326078 (PARI)
%o A326078 a(n)={
%o A326078     my(lim=vector(n, k, sqrtint(k)));
%o A326078     my(accept(b, k)=for(i=2, lim[k], if(k%i ==0 && bittest(b,i) != bittest(b,k/i), return(0))); 1);
%o A326078     my(recurse(k, b)=
%o A326078       my(m=1);
%o A326078       for(j=max(2*k,n\2+1), min(2*k+1,n), if(accept(b,j), m*=2));
%o A326078       k++;
%o A326078       m*if(k > n\2, 1, (self()(k, b) + if(accept(b, k), self()(k, b + (1<<k)))))
%o A326078    );
%o A326078    recurse(1, 0);
%o A326078 } \\ _Andrew Howroyd_, Aug 30 2019
%Y A326078 Cf. A007865, A051026, A054519, A067992, A103580, A325860, A325994, A326023, A326076, A326079, A326081.
%K A326078 nonn
%O A326078 0,3
%A A326078 _Gus Wiseman_, Jun 05 2019
%E A326078 Terms a(21) and beyond from _Andrew Howroyd_, Aug 30 2019