%I #10 Aug 24 2019 21:24:24

%S 1,2,3,6,8,16,24,48,84,168,216,432,648,1296,2448,4896,6528,13056,

%T 19584,39168,77760,155520,229248,458496,790272,1580544,3128832,

%U 6257664,9386496,18772992,24081408,48162816,95938560,191877120,378335232,756670464,1135005696,2270011392

%N Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%C Also the number of subsets of {2...n} with complement containing no term whose prime indices all belong to the subset.

%H Andrew Howroyd, <a href="/A324737/b324737.txt">Table of n, a(n) for n = 1..100</a>

%e The a(1) = 1 through a(6) = 16 subsets:

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

%e {2} {3} {3} {4} {4}

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

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

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

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

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

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

%e {2,3,5}

%e {3,4,5}

%e {3,5,6}

%e {4,5,6}

%e {2,3,4,5}

%e {2,3,5,6}

%e {3,4,5,6}

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

%e An example for n = 15 is {2, 3, 5, 8, 9, 10, 11, 15}. The numbers from 2 to 15 with all prime indices in the subset are {3, 5, 9, 11, 15}, which all belong to the subset, as required.

%t Table[Length[Select[Subsets[Range[2,n]],Function[set,SubsetQ[set,Select[Range[2,n],SubsetQ[set,PrimePi/@First/@FactorInteger[#]]&]]]]],{n,10}]

%o (PARI)

%o pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}

%o a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));

%o ((k, b)->if(k>#p, 1, my(t=self()(k+1, b+(1<<k))); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b), t)); t))(1, 0)} \\ _Andrew Howroyd_, Aug 24 2019

%Y Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844.

%Y Cf. A324697, A324698, A324736, A324738, A324748, A324753, A324755.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 13 2019

%E Terms a(21) and beyond from _Andrew Howroyd_, Aug 24 2019