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%I #9 Aug 27 2019 15:20:17
%S 1,1,2,2,3,4,4,5,6,8,8,11,11,22,22,22,22,28,28,44,44,52,52,76,76,88,
%T 88,96,96,184,184,240,240,264,264,296,296,592,592,592,592,728,728,
%U 1456,1456,1456,1456,2912,2912,3168,3168,3168,3168,5568,5568,5568,5568
%N Number of maximal subsets of {1...n} containing no element 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.
%H Andrew Howroyd, <a href="/A324744/b324744.txt">Table of n, a(n) for n = 0..100</a>
%e The a(1) = 1 through a(8) = 6 maximal subsets:
%e {1} {1} {2} {1,3} {1,3} {1,3,6} {3,4,6} {1,3,6,7}
%e {2} {1,3} {2,4} {1,5} {1,5,6} {1,3,6,7} {1,5,6,7}
%e {3,4} {3,4} {3,4,6} {1,5,6,7} {3,4,6,8}
%e {2,4,5} {2,4,5,6} {2,4,5,6} {3,6,7,8}
%e {2,5,6,7} {2,4,5,6,8}
%e {2,5,6,7,8}
%t maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
%t Table[Length[maxim[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]]],{n,0,10}]
%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, k, if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
%o my(ismax(b)=for(k=1, #p, if(!bittest(b,k) && bitnegimply(p[k], b), my(e=bitor(b, 1<<k), f=0); for(j=k+1, #p, if(bittest(b,j) && !bitnegimply(p[j], e), f=1; break)); if(!f, return(0)) )); 1);
%o my(recurse(k, b)=if(k>#p, ismax(b), my(f=bitnegimply(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<<k)))));
%o recurse(1, 0)} \\ _Andrew Howroyd_, Aug 27 2019
%Y The non-maximal case is A324738. The case for subsets of {2...n} is A324762.
%Y Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A290822, A304360, A306844, A320426, A324764.
%Y Cf. A324694, A324736, A324743, A324749, A324754, A324759.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 15 2019
%E Terms a(16) and beyond from _Andrew Howroyd_, Aug 27 2019
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